Statistical mechanics approach to a general hyperbranched polymer system consisting of ABg monomers and Cf cores

Polymer 55 (2014) 686e697

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Statistical mechanics approach to a general hyperbranched polymer system consisting of ABg monomers and Cf cores Zuo-Fei Zhao a, *, Yuan-Feng Li a, Ning Yao a, Hai-Jun Wang b, c, **, Xin-Wu Ba b a

Faculty of Physics and Electronic Information, Langfang Teachers University, Langfang 065000, PR China College of Chemistry and Environment Science, Hebei University, Baoding 071002, PR China c International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 September 2013 Received in revised form 26 November 2013 Accepted 16 December 2013 Available online 25 December 2013

A general hyperbranched polymerization system that consists of ABg monomers and Cf cores is studied by the principle of statistical mechanics. Two types of partition functions for the polymerization are constructed from different viewpoints, and then the explicit expressions of equilibrium size distribution function, equilibrium free energy as well as the law of mass action are obtained. To investigate the average properties of the system, the number-, weight-average degree of polymerization and polydispersity index are derived. Especially, to investigate the average spatial dimension of the hyperbranched polymers, the mean square radius of gyration is presented as a focus and the influence of unequal reactivity on which is taken into account. As an application, several thermodynamic properties such as the equation of state, isothermal compressibility and specific heat are calculated. In addition, some structural parameters of hyperbranched polymers are also presented under various conditions. The results show that two types of competitions take place during the growing course of hyperbranched polymers, and the reactivity parameter has significant effect on the competitions, which are closely related to the thermodynamic properties and spatial dimension of the hyperbranched polymers. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Free equilibrium energy Unequal reactivity Mean square radius of gyration

1. Introduction In 1952, Flory had proposed that ABg type monomers can be applied to synthesize highly branched polymers, where A and B represent two different functional groups and the subscript g
2 corresponds to the degree of functionality of group B [1]. But before the term of hyperbranched polymers (HBPs) was used, only few relevant experiments were already available during that time. It was until Webster and Kim realized HBPs by one-pot synthesis that a considerable attention had been attracted due to the unique properties of such polymers [2e8]. Compared with linear analogs, the HBPs exhibit unique properties such as without entanglement, low viscosity, high solubility, high degree of functionality, globular architecture and good capacity of encapsulating guest molecules [9e11]. Hence the HBPs have many potential applications in polymer science and engineering. Up date, more and more researchers

* Corresponding author. Faculty of Physics and Electronic Information, Langfang Teachers University, Langfang 065000, PR China. ** Corresponding author. College of Chemistry and Environment Science, Hebei University, Baoding 071002, PR China. E-mail addresses: [email protected] (Z.-F. Zhao), [email protected] (H.-J. Wang). 0032-3861/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.polymer.2013.12.036

focus on some new synthesis strategies and try to extend such polymers to regimes of coatings, additives, nanotechnology, chemical biology and so on [9e15]. Theoretically, a series of insight investigations on HBPs have been performed through different methods. Of which, by means of the probabilistic analysis, Flory gave the molecular size distribution function of polymers formed by ABg type monomers [1]. Fréchet et al. introduced an important polymeric parameter, the degree of branching (DB), to characterize the structural property of HBPs [16]. Möller et al. presented the expression of DB of ABg system [17]. For such a system, Ba and coworkers got the explicit expressions of zaverage mean square radii of gyration (MSRG) and discussed its scaling behaviors [18,19]. To consider intramolecular cyclization, a computer simulation based on the lattice model was used to simulate AB2 system by Fawcett [20] and a bridge function was used to investigate ABg system by Dusek [21]. Then the ABg (g ¼ 2, 4) systems were also studied by He and coworkers through an extended 3D bond fluctuation lattice model, where the intramolecular cyclization, diffusion of monomers and molecules, relaxation of polymer chains, monomer concentration, reaction rate constant and substitution effect were considered [22e24]. Recently, Gray-Weale group applied a mean field model to deal with the interactions between different parts of the polymer by

Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

FloryeHuggins-like treatment, and the thermodynamic properties as well as the MSRG of HBPs are simulated [25,26]. The great amount of experimental and theoretical works showed that the resultant HBPs from pure monomers often have a wide size distribution [9e11]. To improve the polydispersity of HBPs, the addition of multifunctional core moieties is explored. And the corresponding binary polycondensation systems such as multifunctional cores with AB, AB2 or ABg monomers have been studied by Yan and Zhou through solving kinetic differential equations [27e32]. In their studies, the explicit expressions of size distribution function, the number-, weight- and z-average molecular weight as well as polydispersity are obtained. In addition, Ba and coworkers studied the conversion dependence of the MSRG of star-branched polymers made by AB þ Af type polycondensation and investigated the mean size distribution of HBPs in an AB2 þ Bf system [33]. Frey and coworkers simulated the slow addition of monomers to multifunctional cores [34]. The binary systems mentioned above were all performed based on the assumption of equal reactivity. In practice, the reactivity difference between cores and monomers has significant influences on the mean dimension and properties of HBPs [9e11]. For this purpose, Cheng made use of a method of generating function to investigate an AB2 monomers and C3 cores system and the influence of reactivities on the degree of polymerization (DP), DB and polydispersity were discussed [35]. Recently, a system consisting AB2 monomers and Cf cores has been investigated by Zhou and Yan through solving kinetic differential equation method, and the influence of different reactivity on the molecular size distribution, number-, weight-average molecular weight and polydispersity have been expressed in analytical forms [36]. In this paper, a more general unequal reactivity system that consists of ABg type monomers and Cf (subscript f
3 denotes the degree of functionality) core molecules is studied by the method of statistical mechanics [37e41], and the influences of unequal reactivity on the average polymeric quantities are considered. For the system under study, HBPs with various sizes and different configurations can form due to polymerization. Hence the method of statistical mechanics can be served as a powerful tool to treat the relevant physical and chemical properties of the system. It is obvious that, both groups B and C can react with group A but with different reactivity. As a result, two types of HBPs can form due to the presence of core molecules, one type is HBPs with a core, the other is HBPs without a core. Then as an application, the effect of unequal reactivity on the equilibrium free energy, law of mass action, size distribution function, number-, weight-average DP and the polydispersity index are studied separately and wholly, respectively. In particular, the MSRG is emphasized to characterize the average spatial dimension of the two types of HBPs. In addition, several thermodynamic properties and some structural parameters of HBPs are calculated as well. For easy of presentation, hereafter the physical quantities for the HBPs with a core will be marked with a superscript “c” to distinguish them from those without a core. In addition, throughout the paper, the intramolecular cyclization is excluded. The remainder of this paper is organized as follows. In Section 2, a method of statistical mechanics is applied to study the present system. Two types of partition functions are constructed from different viewpoints and then the equilibrium free energy, law of mass action as well as equilibrium size distribution functions are carried out. In Section 3, the k-th moment of HBPs and its application are discussed. In detail, based on the definition of k-th moment, two recursion formulas satisfied by HBPs with and without a core are derived. Then starting from a selected starting point, the first and second moments of different HBPs are explicitly calculated. As an application, the number-, weight-average DP as

687

well as the polydispersity of HBPs are presented. In Section 4, in terms of Kramers theorem, the recursion formula of MSRG is derived and the properties of MSRG are investigated under various conditions. The results show that related factors such as the reactivity parameter, stoichiometric parameter and functionalities have significant influences on the average spatial dimension of HBPs. In Section 5, as an illustration, the equation of state, isothermal compressibility and specific heat concerning polymerization are presented based on the free energy, meanwhile the structural parameters of HBPs are investigated, which refer to the numbers of monomers, terminal units, chain units and branched units. In Section 6, a summarization is made and some related problems and limitations of this paper are discussed. 2. Application of statistical mechanics in the present system To begin, a binary system that consists of M cores of Cf type and N monomers of ABg type is introduced, where the reactions only take place between groups A and B or A and C, and as a result two types of bonds TAB and TAC are generated in the system. At a given time, one can assume the number of TAB type bonds is LB and that of TAC is LC. Thus the number of reacted group A, denoted by L, should satisfy the relation that L ¼ LB þ LC. Due to polymerization, HBPs with different sizes can form through the connection of TAB and TAC bonds. Accordingly, the partition function can be constructed from two different viewpoints. One is based on the consideration of polymeric configurations, the other is related to the ways of bonding from functional groups. In the following, two types of partition functions are constructed based on the reference state at which there are no bonds generated. 2.1. Partition function from the viewpoint of bonding, the law of mass action and equilibrium free energy From the viewpoint of bonding, a partition function of the system can be constructed as

Q1 ðM; N; LB ; LC Þ ¼ C1 ðM; N; LB ; LC Þ½KB ðTÞLB ½KC ðTÞLC

(1)

where KB(T) and KC(T) are both associated with the absolute temperature T, and denote the relative probabilities of generating a TAB or TAC type bond, respectively. The combinational factor C1(M, N, LB, LC) represents the number of ways of forming LB and LC bonds by the M cores of Cf and N monomers of ABg, it takes the form as

C1 ðM; N; LB ; LC Þ ¼

N! ðgNÞ! ðfMÞ! LB !LC !ðN  LÞ! ðgN  LB Þ! ðfM  LC Þ!

(2)

As for KB(T) and KC(T) in Eq. (1), they are closely related to thermodynamic conditions and bonding energies of εB and εC, respectively. To show their physical meanings more clearly, one can express them as the following forms

vB ½expðbεB Þ  1 V v KC ðTÞ ¼ C ½expðbεC Þ  1 V

KB ðTÞ ¼

(3)

where b1 h kBT with the Boltzmann constant kB, and vB(vC) is the bonding volume of TAB(TAC) type bond. The above equations indicate that to generate a new bond, two conditions must be satisfied simultaneously: the former is that the two groups of A and B (or A and C) must enter into a bonding volume vB (or vC), the latter is that the change in energy εB (or εC) must also satisfy the corresponding bonding requirement. It should be noted that the initial state of L ¼ 0 has been considered in the above equations.

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By virtue of the partition function in Eq. (1), the free energy can be expressed as a function of LB and LC

F1 ðLB ; LC Þ ¼ b

1

½LB ln KB ðTÞ þ LC ln KC ðTÞ

þ ln C1 ðM; N; LB ; LC Þ

(4)

Minimizing the free energy F1(LB, LC) with respect to LB and LC, respectively, KB(T) and KC(T) can also be derived as

KB ðTÞ ¼ KC ðTÞ ¼

lB

ðN  lÞðgN  lB Þ

(5)

lC

ðN  lÞðfM  lC Þ

where l, lB and lC correspond to the equilibrium values of L, LB and LC. The above expressions are substantially the law of mass action and can be regarded as the corresponding equilibrium constants of forming bonds of TAB and TAC type, respectively. Meanwhile, combining Eq. (4) with Eq. (5) together, the equilibrium free energy Feq(N, l) can be carried out

Feq ðN; lÞ ¼ b

1

n





l þ N ln ð1  xÞð1  xB Þg þ M lnð1  xC Þf

o

(6)

In the expression, x, xB and xC are the conversions of groups A, B and C, which have the definitions of x ¼ l=N, xB ¼ lB =ðgNÞ and xC ¼ lC =ðfMÞ, respectively. Furthermore, if a stoichiometric parameter r is introduced as the molar ratio of Cf cores to ABg monomers, namely, r ¼ M=N, the three conversions satisfy the following relation

x ¼ gxB þ rfxC

(7)

2.2. Partition function from the viewpoint of polymers and equilibrium size distribution function As a consequence of the polymerization, polymers with various sizes appear in the system, and then we can also construct partition function from viewpoint of polymers. To this end, with the aid of statistical method used by Stockmayer [42], it can be written as

Q2 ðM; N; LB ; LC Þ ¼ C2 ðM; N; LB ; LC Þ½KB ðTÞLB ½KC ðTÞLC

(8)

in which KB(T) and KC(T) have the same physical meanings as those in Eq. (1), and the factor C2(M, N, LB, LC) represents all the possible arrangements of ABg monomers and Cf cores. The detailed form of C2(M, N, LB, LC) can be given by

C2 ðM; N; LB ; LC Þ ¼ M!N!

h iP c i;m Y uPi uci;m i

i;m

Pi !

(9)

c ! Pi;m

c where Pi denotes the number of i-mers without a core, and Pi;m denotes the number of i-mers with a core, of which there are m reacted C groups, while ui and uci;m represent the corresponding c obey the redegeneracies. For the system under study, Pi and Pi;m strictions that

M ¼

X i;m

LB ¼

c Pi;m ; N ¼

X i

iPi þ

X c ði  1ÞPi;m

(10)

i;m

X X X c c ði  1ÞPi þ ði  m  1ÞPi;m ; LC ¼ mPi;m i

i;m

(11)

i;m

Expressions in Eq. (10) are virtually the material balance conditions, which mean that the two types of structural units are

conservative in the polymerization, while Eq. (11) stems from the assumption that intramolecular cyclizations are excluded. In fact, the degeneracies of ui and uci;m can be derived from a probabilistic consideration. According to the assumption of without intramolecular cyclization, for an i-mer without a core, it needs i monomers of ABg type to be connected by (i  1) bonds of TAB type. Hence the expression of ui can be straightforwardly given by ui ¼ ðigÞ!=fi!½iðg  1Þ þ 1!g. For an (i þ 1)-mer with a Cf core that consists of m reacted groups of C, there are f =i ways for it to connect with the i monomers of ABg, and ðf  1Þ!=½ðm  1Þ!ðf  mÞ! ways for selecting the residual (m  1) groups of C from other (f  1) ones to anticipate the reactions between groups C and A. On the other hand, the different bonding ways for the (i  m) bonds of TAB type should be ðigÞ!=fði  mÞ!½ðg  1Þi þ m!g, hence the expression of uciþ1;m can be obtained as uciþ1;m ¼ f !=fðf  mÞ!ðm  1Þ!ðigÞ!= iði  mÞ!½ðg  1Þi þ m!g. Accordingly, for such an i-mer with a Cf core, uci;m can be directly given out through the replacement of (i þ 1) to i in the derivation. So far, from different viewpoints, two forms of partition functions have been constructed for the same system over the same reference state. They can be proved to be consistent with each other and can be further applied to obtain some thermodynamic quantities of the system by the principle of statistical mechanics. In terms of the partition function in Eq. (8), the free energy can also be expressed as 1

F2 ðLB ;LC Þ ¼ b

½LB ln KB ðTÞþLC ln KC ðTÞþln C2 ðM;N;LB ;LC Þ (12)

Obviously, two free energy expressions of F1(LB, LC) and F2(LB, LC) should be the same minimum in equilibrium state. Thus upon comparing Eq. (4) with Eq. (12), one may find that C1(M, N, LB, LC) should equal to C2(M, N, LB, LC). Then differentiating both sides of equation C1(M, N, LB, LC) ¼ C2(M, N, LB, LC) with respect to Pi and c under the restrictions of Eqs. (10) and (11), the equilibrium size Pi;m c can be given by distribution functions (ESDFs) of HBPs Pi and Pi;m

Pi ¼ N ui ð1xÞðxB Þi1 ð1xB Þðg1Þiþ1 c ¼ M uc ðx Þim1 ðx Þm ð1x Þði1Þðg1Þþm ð1x Þf m Pi;m B C C i;m B

(13)

Furthermore, the above ESDFs can also be rewritten as

Pi ¼ ui ðP1 Þi ½KB ðTÞi1 c c ðP Þi1 ½K ðTÞim1 ½K ðTÞm Pi;m ¼ uci;m P1;0 B 1 C

(14)

c where P1 ¼ Nð1  xÞð1  xB Þg and P1;0 ¼ Mð1  xC Þf denote the numbers of unreacted ABg monomers and free Cf cores in equilibrium state. It should be noted that when the degree of functionality g is equal to 2, the results in Eq. (13) are entirely identical to those obtained by the method of solving kinetic differential equations, which have been performed by Zhou and his coworkers [36]. Furthermore, the expressions in Eq. (14) reflect the fact that it needs i monomers react (i  1) times to form an i-mer. In particular, to form an i-mer with a core, a Cf core and (i  1) monomers are needed to react (i  1) times, of which (i  m  1) times with the probability of KB(T) to generate a TAB bond, and m times with the probability KC(T) for a TAC bond. These results agree well with the theory of reaction dynamics for polymerization [43]. As expressed in Eq. (3), KB(T) and KC(T) are closely related to thermodynamic conditions and bonding energies of εB and εC. Therefore several thermodynamic properties and some structural information of the HBPs can be obtained in terms of the present statistical mechanics method. In so doing, the relationships between some average polymer quantities and thermodynamic

Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

conditions can be found. The above analysis means that the two different forms of partition functions are consistent with each other, and hence the relevant results can be further applied to investigate the physical and chemical properties of the system. 3. The k-Th moment of HBPS and its application Besides ESDFs, the polydispersity index (PI) is another important polymeric quantity to characterize the distribution of HBPs with different DP. So far, it has attracted great interests in numerous experimental and theoretical works [9e11]. Theoretically, PI is closely related to the number- and weight-average DP [28e32]. For this purpose, the k-th polymeric moments of HBPs with and P k c P k without a core are introduced as Mkc ¼ i Pi;m and Mk ¼ i Pi , i;m

i

respectively. According to this definition, a recursion formula about moment can be derived then the higher moments can be calculated provided that a starting point is given. 3.1. The k-th moment of HBPs and its recursion formula c in Eq. (13) into definitions of Substituting the ESDFs of Pi and Pi;m the k-th moments Mk and Mkc , respectively, and differentiating both sides of the two equations with respect to conversions of xB and xC, recursion formulas satisfied by the two types of HBPs can be obtained as

Mkþ1 ¼

c Mkþ1

  xB ð1  xB Þ vMk gx ð1  xB Þ Mk þ 1þ B 1  gxB vxB 1x 1  gxB

  x ð1  xB Þ vMkc xC ð1  xC Þ vMkc fxC Mkc ¼ B þ þ 1þ 1  gxB vxB 1  gxB vxC 1  gxB

(15)

M1 ¼ M2 ¼

Nð1  xÞ 1  gxB

i h Nð1  xÞ 1  gðxB Þ2

The expression of M1c means that, in average, each Cf core will connect with fxC =ð1  gxB Þ molecules of ABg type. In the relevant calculations, the relation of x ¼ gxB þ rfxC has been used. Based on the above mentioned polymeric moments, the k-th moments of the whole system Mks can be obtained according to the formula that Mks ¼ Mkc þ Mk . Hence the zeroth, first and second moments of the whole system can be written as

M0s ¼ Nð1  xÞ þ M M1s ¼ N þ M i h i h N 1  gðxB Þ2 þ M ð1  gxB þ fxC Þ2  f ðxC Þ2 M2s ¼ ð1  gxB Þ2

(17)

ð1  gxB Þ3

For HBPs with a core, its zeroth moment M0c denotes the number of HBPs with a core in the system and should equal to the initial P c Pi;m . Then number of the core molecules, namely, M0c ¼ M ¼ i;m

selecting it as a starting point, and successively using the recursion formula in Eq. (16), the first and second moments can be given by

fxC M 1  gxB h M fx gx ð1  xB Þi fxC ð1  xC Þ þ ð1  gxB þ fxC Þ2 þ C B M2c ¼ 2 1  gxB ð1  gxB Þ

 M1c ¼ 1 þ

(18)

(19)

So far, the zeroth, first and second polymeric moments of two types of HBPs and that of the whole system have been obtained, and they can be further used to calculate some physical quantities such as the number-, weight-average DP and PI.

3.2. The number-, weight-average degree of polymerization and the polydispersity index As is well known, the number- and weight-average molecular weight are two important quantities to characterize the average size of HBPs. They are closely related to the DP, and can be easily calculated by the above moments. As a result, for HBPs without a core, the number- and weight-average DP, DPn and DPw, can be obtained as

DPn ¼

M1 M0

¼

1 1gxB

DPw ¼

M2 M1

¼

1g ðxB Þ2 ð1gxB Þ2

(16)

The recursion formula indicates that once the k-th moment is given, the (k þ 1)-th moment can be derived only through a differential calculation provided a starting point is given. For HBPs without a core, the zeroth moment M0 means the number of such kind of polymers, therefore it can be given straightforwardly by this physical meaning. For the present system, the number of HBPs without a core is just equal to the number of free groups A, and can be expressed as M0 ¼ N(1  x). Thus one can take it as a starting point to give the higher order moments according to the recursion formula in Eq. (15). As an illustration, the first and second moments are obtained as

689

(20)

Similarly, for HBPs with a core, the number- and weight-average c , can be derived as DP, DPnc and DPw

DPnc ¼

M1c M0c

c ¼ DPw

M2c M1c

fxC ¼ 1 þ 1gx B

h i 1xC þgxB ðxC xB Þ fxC ¼ 1 þ 1gx 1 þ ð1gx Þð1gx þfx Þ B B

B

(21)

C

For the whole system, the number- and weight-average DP, DPns s , can be given by and DPw

DPns ¼

M1s M0s

¼

s ¼ DPw

M2s M1s

¼

1þr 1þrx 1g ðxB Þ2 þr ½ð1gxB þfxC Þ f ðxC Þ2  2

(22)

ð1þr Þð1gxB Þ2

Now the expressions of number- and weight-average DP for the two types of HBPs and the whole system have been explicitly presented, they can be further applied to study the PI. In practice, PI is a fundamental polymeric quantity to measure the degree of uniformity in polymeric sizes and has caused great attentions in many experimental and theoretical works [9,10]. According to its definition, the PIs of HBPs without a core PI, with a core PIc and that of the whole system PIs can be given as 1g ðxB Þ w PI ¼ DP DPn ¼ 1gxB

2

PI c ¼ DPwc ¼ 1 þ fxC ½1xC þgxB ðxC xB Þ2 n ð1gxB Þð1gxB þfxC Þ

 2 ð1þrxÞ 1gðxB Þ2 þr½ð1gxB þfxC Þ f ðxC Þ2  DP s PI s ¼ DPws ¼ 2 2 DP c

n

ð1þrÞ ð1gxB Þ

(23)

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Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

It should be noted that, when g equals to 2, the polymeric moments, DPs and PIs are all identical to those obtained by Zhou and his coworkers [36]. So far, by the method of statistical mechanics, not only the ESDFs in Eqs. (13) and (14) but also the polymeric moments, DPs and PIs are derived in their correct results, this signifies that the present statistical mechanics method is credible and can be further applied to make an insight study on the present system. To investigate the influences of different reactivities on the quantities obtained above, the relation between xB and xC can be given by the following differential kinetic equations.

P dNB ¼ kAB NB Pi dt i P dNC ¼ kAC NC Pi dt i

(24)

where NB(NC) denotes the number of unreacted groups B(C) in the present system, and kAB(kAC) represents the reaction rate constant of groups B(C) with A. Obviously, the above equations describe the consumption of groups B and C, where the summation of Pi is essentially the number of free groups A. For the system under study, both groups B and C will react with free groups A. Hence there exists a competition between the consumption of groups B and C, which can be reflected from the different reaction rate constants kAB and kAC. Experimentally, the effect of this unequal reactivity on the structure and property of resultant HBPs is obvious [9e11]. Theoretically, to regulate the interesting physical quantities by changing reactivity, a reactivity parameter q can be introduced as

q ¼

kAB kAC

(25)

Then consider the definitions that of xB, xC and the zeroth moment of polymers without a core, the relation between xB and xC can be obtained as

xC ¼ 1  ð1  xB Þ1=q

(26)

Making use of this equation together with Eq. (7), the conversion of groups A can be expressed as a function of xB and q

i h x ¼ gxB þ fr 1  ð1  xB Þ1=q

s against x with r ¼ 0.01, f ¼ 3 and g ¼ 3 for different reactivity Fig. 1. The plots of DPw parameters q ¼ 0.1, 1, 10 and 100.

4. The mean square radius of gyration and its recursion formula As is well known, the MSRG plays a fundamental role in characterizing the dimension of branched polymers [18,19,25,33,44e 49]. Experimentally, the MSRG can be detected by the method of small angle laser light scattering [44]. Theoretically, besides the computer simulation method [45], analytical techniques such as statistical method [46], stochastic theory of branching processes [47] and kinetic approach method [48] have been applied to study the MSRG. Up to date, all these successful studies provide us important information for understanding the MSRG. Nevertheless, works about the effect of the unequal reactivity on MSRG are seldom reported. In this paper, the unequal reactivity will be taken into account in the calculation of MSRG, as shown below. 4.1. The explicit form of the second MSRG In general, to characterize the average dimension of HBPs in a system, the k-th (k ¼ 0, 1, 2.) MSRG is often introduced [18,19,39e c 42,48,49]. For the present system, hR2 ik and hR2 ik are applied to represent the k-th (k ¼ 0, 1, 2.) MSRGs of polymers without a core and those with a core, respectively, they are the averaged MSRGs over all species of polymers in such a way that

(27)

Combining Eqs. (26) and (27) with the expressions of number-, weight-average DP and PI, the influences of reactivity on the corresponding quantities can be investigated straightforwardly. s and PIs As an example, Fig. 1 and 2 present the plots of DPw versus x under the conditions that f ¼ g ¼ 3 and r ¼ 0.01. To learn about the effect of unequal reactivity, the reactivity parameter q has been taken as 0.1, 1, 10 and 100, respectively. Seen from Fig. 1, one can find that, when the reactivity parameter q varies from 0.1 to s changes apparently with the increase in x, and its 100, DPw magnitude almost runs over six orders near the end of polymerization. Similarly, Fig. 2 shows that the PIs also changes about five orders in its magnitude. These results indicate that the reactivity parameter q does have significant influences on the mean size of s and a wide mean size HBPs, and a large q leads to a large DPw distribution. Therefore, the effect of unequal reactivity on other average polymeric quantities can be further anticipated. As an illustration, below, the MSRG will be demonstrated to investigate the spatial dimension of HBPs.

Fig. 2. The plots of PIs against x under the same conditions as that of Fig. 1.

Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

691

In terms of Kramers theorem, each polymer can be divided into two moieties by cutting any a TAB or TAC type bond imaginarily [18,19,38e40], hence hR2i i and hR2i;m ic can be expressed as

D E R2i ¼

b2 i2

D Ec R2i;m ¼

P

i0 ði  i0 ÞDði; i0 Þ

i0 b2 i2

P

(29) i0 ði  i0 ÞDc ði; i0 Þ

i0 0

where b represents the Kuhn bond length, [49] D(i, i ) denotes the all possible ways of decomposing each i-mer without a core into an 0 0 0 i -mer and an (i  i )-mer, while Dc(i, i ) denotes the all possible 0 ways of splitting an i-mer with a core into an i -mer with a core and 0 0 an (i  i )-mer without a core. To calculate the k-th MSRG, D(i, i ) and 0 c c . For easy of D (i, i ) are often expressed as functions of Pi and Pi;m c presentation, the results of the second MSRGs, hR2 i2 and hR2 i2 , are directly written as below, and the detailed derivations are given in the Appendix. Fig. 3. The plots of hR2 i2 against x with f ¼ 3, g ¼ 3 and r ¼ 0.01 at different reactivity parameters q ¼ 0.01, 1, and 100.

2 P kD 2E i Ri Pi R k ¼ i 2 c P k D 2 Ec c i Ri;m Pi;m R k ¼

2 c R 2 ¼

Mb2 fxC h ð1  gxB Þ3

(30)

gðg  1ÞðxB Þ2 i 1  xC þ fxC þ 1  gxB c

(28)

i;m

where hR2i i represents the ensemble average of MSRG for an i-mer without a core over its all possible configurations, and hR2i;m ic that of an i-mer with a Cf core in which there are m reacted C groups. They can be evaluated by the Kramers theorem, which relates the size of polymers to a purely structural property and has developed into several forms [46e49]. In essence, the MSRG of a polymer involves the summation of each basic unit. To this end, a polymer is often imaginarily split into any two possible parts. Through this operation, a relation between basic units and bonds can be found. Then instead of the summation over structural units, the summation over bonds can be performed [49]. By this transformation, the Kramers theorem can be expressed by the form that average over all possible ways of dividing a polymer into two moieties. In practice, combining the Kramers theorem and the definition of k-th MSRG together, some certain MSRG can be calculated. As an application, the second MSRG of HBPs will be demonstrated.

c

2 Nb2 gxB ð1  xÞð1  xB Þ R 2 ¼ ð1  gxB Þ4

Fig. 4. The plots of hR2 i2 against x under the same conditions as those in Fig. 3.

The above equations indicate that hR2 i2 and hR2 i2 are explicit functions of conversions x, xB and xC. Because x, xB and xC can be related by the reactivity parameter q (see Eq. (26) and Eq. (27).), the c second MSRGs of hR2 i2 and hR2 i2 can be further used to study the effect of reactivity parameter on the average spatial dimension properties of HBPs. 4.2. The effect of some factors on the MSRG To investigate the properties of MSRG of the system, the plots of c hR2 i2 and hR2 i2 in unit of Nb2 against x under various conditions are presented in Figs. 3e10. In detail, the effects of different factors on the average dimension of two types of HBPs are focused on four aspects, they are the reactivity parameter q in Figs. 3 and 4, the stoichiometric parameter r in Figs. 5 and 6, the degree of core molecule functionality f in Figs. 7 and 8, and the degree of functionality g in Figs. 9 and 10. The results show that the MSRG of HBPs in the present system have these properties: (i). With an increase in the reactivity parameter q, hR2 i2 inc creases as shown in Fig. 3, while hR2 i2 firstly decreases in a very wide range of x, then an inversion appears near the end of polymerization. This can be interpreted as follows. For the present system, there exists a competition between groups B and C, a large reactivity parameter q means that the free functional groups A prefer to react with groups B rather than groups C. Thus a rapid increase of hR2 i2 and a smooth inc crease of hR2 i2 can be observed at the initial reaction stage. But at the last stage of polymerization, reactions mainly take pace between two types of HBPs, the combination of large size HBPs without a core will lead to the rapid increase of HBPs with a core, and the inversion occurs, as shown in Fig. 4. (ii). With an increase in the stoichiometric parameter r, hR2 i2 c decreases monotonously as shown in Fig. 5, while hR2 i2 firstly increases with r, then it inverts at higher conversion as shown in the inside panel of Fig. 6. In essence, the increase in r means there are more functional groups C in the system. It is easy to be understood, besides the reactivity, the relative number of groups B and C is another important factor which affects the competition between groups B and C. Obviously,

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Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

Fig. 5. The plots of hR2 i2 against x with f ¼ 3, g ¼ 3 and q ¼ 10 at different molar ratios r ¼ 0.0001, 0.001 and 0.01.

the more functional groups C is in favor of its competition against functional groups B at the initial reaction stage. However, gradually, and in particular after a certain conversion, a competition between two types of HBPs dominates due to the shortage of free functional groups A. As a result, the growth of HBPs with a core will mainly depend on the mean dimension of the combined HBPs without a core. Combining the two types of competitions during different reaction stages together, the decrease of hR2 i2 and the c inversion of hR2 i2 can be understood. Similar to the role of r, a large degree of functionality f also signifies more functional groups C. And therefore with the increase in f, the plots of hR2 i2 in Fig. 7 display the same tendency as those in Fig. 5, c and the tendency of hR2 i2 in Fig. 8 is similar to that in Fig. 6. The only difference between r and f is that the increase of r means more Cf cores are introduced, while the increase of f does not change the number of core molecules. (iii). With an increase in the degree of functionality g, hR2 i2 inc creases monotonously as shown in Fig. 9, while hR2 i2 undergoes an inversion, namely, it decreases with the increase of g at a low conversion, and then increases at a high conversion as shown in Fig. 10. Obviously, the increase of g means that more functional groups B will be introduced, which is in favor of the growth of HBPs without a core. Then

c

Fig. 6. The plots of hR2 i2 against x under the same conditions as those in Fig. 5.

Fig. 7. The plots of hR2 i2 against x with g ¼ 3, q ¼ 10 and r ¼ 0.01 at different degree of functionalities f ¼ 2, 3, 4.

considering the two types of competitions during different reaction stages as mentioned above, the tendencies of hR2 i2 c in Fig. 9 and hR2 i2 in Fig. 10 can be understood easily. Collecting all the results shown in Figs. 3e10, one can find that, c with an increase in conversion x, hR2 i2 increases monotonously, while hR2 i2 firstly increases then decreases, and each of them goes through a maximum. This common feature coincides with the reaction process very well. For the present system, the growth of HBPs is at the cost of consuming free functional groups A. At the initial stage of polymerization, the great amount of unreacted groups A can c ensure the HBPs with and without a core to grow freely, thus hR2 i2 2 and hR i2 both increase with the increase in conversion x. But at the last reaction stage, reactions mainly take place between the two types of HBPs due to the shortage of unreacted groups A. As a result, the combination of HBPs without a core into HBPs with a core leads c to the continuous increase of hR2 i2 . The maximum of hR2 i2 indicates that in average, the largest dimension of HBPs without a core appears only when the conversion is over a certain value which is determined by the parameters of q, r, f and g, below which one can think that HBPs with and without a core can grow freely by themselves and above which, the reactions mainly take place between

c

Fig. 8. The plots of hR2 i2 against x under the same conditions as those in Fig. 7.

Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

693

show apparent changes. Chemically, the polymerization can be characterized by the number- and weight-average DP as well as the MSRG. Interestingly, one can also investigate the effect of polymerization on the system from a physical viewpoint even though an ideal model of mixtures is taken. As an application of the present method, the major aim in this section is twofold. On the one hand, we will start from equilibrium free energy and the law of mass action to give several thermodynamic properties of the system such as the equation of state, isothermal compressibility and specific heat. On the other hand, it is also interesting to consider how the structural parameters concerning the two types of HBPs to vary with different reaction conditions, and the molecular structural parameters such as the numbers of monomers, terminal units, chain units and branched units are also calculated. 5.1. Several thermodynamic properties of the system Fig. 9. The plots of hR2 i2 against x with f ¼ 3, q ¼ 1 and r ¼ 0.01 at different degree of functionalities g ¼ 1, 2, 3 and 4.

two types of polymers. On the other hand, the maximum of hR2 i2 has a shift with conversion x. A large q or g makes the maximum delay with the conversion x, while a large r or f has an opposite effect. In fact, the reactivity parameter q reflects the effect of unequal reactivity between groups B and C, while the parameters of r, f and g change their relative number. The different reactivity and distinct number between groups B and C indicate that there exists a competition among them. This competition leads to significant effect on the mean dimension of HBPs and can be controlled by designing suitable parameters such as q, r, f and g. Especially, the reactivity parameter q may vary in a very wide range and plays an important role in regulating the mean dimension of resultant HBPs, which offers a possible strategy to synthesize materials with desired structure and properties. 5. Several thermodynamic properties of the system and some structural parameters of the HBPS In the preceding sections, we have studied several average properties of the system resulted from the polymerization. Clearly, it is the polymerization that results in the change in connectivity between molecules such that some relevant physical quantities also

c

Fig. 10. The plots of hR2 i2 against x under the same conditions as those in Fig. 9.

For an ideal model of mixtures, any of monomers, core moieties and polymers can be thought of as a mass point when the excluded volume interactions have been neglected. Under such an assumption and making use of the equilibrium free energy, several thermodynamic quantities can be further used to describe the changes caused by polymerization in the system. Physically, this would provide us new insights on the polymerization. As a thermodynamic quantity, the change of pressure P is so apparent and can be directly reflected from the equation of state. For the initial state of the system under study, the equation of state can be written as PV ¼ b1(M þ N), where an ideal model of mixtures has been used. When the polymerization takes place, molecules connect one by one and thus one can expect the corresponding change in the pressure. Eventually, only M polymers are left in the system, hence we have PV ¼ b1M, which is independent of the intra-molecular cyclization being excluded. In fact, the equation of state can be derived from the equilibrium free energy by the formula P ¼ vFeq =vV. Substituting Feq in Eq. (6) into this formula and combining the law of mass action in Eqs. (3) and (5), one can find that the pressure P satisfies the following equation 1 PV ¼ b ðM þ N  lÞ

(31)

This equation of state indicates that the pressure of system is closely related to the polymerization. In the initial stage of polymerization, the equation of state degenerates to PV ¼ b1(M þ N) because of l ¼ 0. Whereas, eventually, all the N monomers of ABg type participate in the reactions and the number of new formed bonds becomes l ¼ N, thus the equation of state becomes PV ¼ b1M. This agrees well with our above analysis. From a chemical viewpoint, it is the polymerization that results in the formation of HBPs, and experimentally, which is usually characterized by the measurements of the number- and weightaverage molecular weight, radius of gyration, viscosity and so on. Physically, polymerization means that through the connection of bonds, the spatial correlation between structural units is increased with an increase in the conversion. And quite appropriately, there exists a physical quantity to characterize such a spatial correlation and, theoretically, it is the isothermal compressibility of the system. With the present statistical mechanics method, the isothermal compressibility cT is closely related to the weight-average DP, as will be shown below. Making use of the equation of state together with the law of mass action given by Eqs. (3) and (5), the isothermal compressibility (defined as cT ¼ ð1=V ÞvV=vPj:T ) can be carried out

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Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

cT s ¼ DPw cid T

(32)

where cid T ¼ bV=ðM þ NÞ is the isothermal compressibility in the s denotes the weight average DP given in Eq. (22). ideal state and DPw As is well known, the isothermal compressibility measures the structural factor of a system in the long wavelength limit, which is closely related to the molecular number density fluctuation [39]. Clearly, with an increase in the conversion x, more and more monomers link together and as a result, the average connectivity between molecules increases because of the presence of new generated bonds. Therefore one can know that, the molecular number density must decrease gradually because of the polymerization, and accordingly, the spatial correlation of the structural units in the system increases. Note that, the left hand side of Eq. (32) is a thermodynamic quantity, while its right hand side is the weight-average DP, a pure geometrical quantity. Since the average dimension of the HBPs can be measured by a light scattering experiment (e.g. MSRG), where the structure factor plays an important role. This indicates that the structural properties and thermodynamic properties of a polymerization system are always relevant. Recalling that, in essence, the isothermal compressibility reflects the fluctuationecorrelation property of a system, thus polymerization between molecules also provides an example to justify such a correlation through bonds. Seen from Fig. 1, it can be s increases observed that with the increase in conversion x, DPw monotonously. This means that the correlation between structural units increases with the conversion x. In addition, in terms of the equilibrium free energy and the law of mass action, specific heat Cv can also be evaluated. To this end, it is necessary to give the change in internal energy DU. Applying the   relation that DU ¼ v bDFeq =vb, we have

DU ¼ lB εB  lC εC

(33)

Clearly, this result reflects the change in internal energy owing to polymerization. Thus the change in specific heat DCv caused by polymerization can be calculated by DCv ¼ vDU=vTjv and takes the form

from polymerization. As an example, the equation of state, the isothermal compressibility and the specific heat have been calculated and the corresponding physical meanings have been interpreted. Along this clue, the further investigations on some other thermodynamic quantities are expected. 5.2. The effect of reaction conditions on the structural parameters of HBPs From a physical viewpoint, the correlation resulted from polymerization is caused by the connection between monomers. Due to the polymerization, monomers will turn into different molecular structural units. In general, according to the number of reacted functional groups, molecular structural units can be classified as the following four types: (1) Monomer. This refers to a monomer whose groups do not participate in any reactions; (2) Terminal unit. This refers to a monomer that only one of its groups has reacted; (3) Chain unit. This refers to a monomer that only two of its groups participate in the reactions; (4) Branched unit. This refers to a monomer that three or more of its groups have reacted. In practice, the numbers of these four quantities are closely related to the molecular structures of HBPs, which are important for both experimental and theoretical works. Experimentally, the NMR technique has been applied to characterize HBPs due to the chemical shifts of different molecular structural units are distinct from one another, and some useful information on the molecular structures of HBPs has been obtained [50]. Theoretically, the numbers of these distinct structural units can be calculated in a straightforward manner, and can be expressed as an explicit function of conversion. In this subsection, to investigate the influences of some parameters such as q, r, f and g on the molecular structures of HBPs, the numbers of four types of structural units mentioned above will be calculated and discussed as follows. Recalling that x, xB and xC are the conversions of functional groups A, B and C, respectively, they are essentially the probabilities of finding the corresponding group to participate in the reaction. In terms of these quantities, the number of molecular structural units can be calculated. If the numbers of monomers, terminal units, chain units and branched units are in turn denoted by Nm., Nt.u., Nc.u. and Nb.u., they can be obtained as follows

Nm: ¼ Nð1  xÞð1  xB Þg þ Mð1  xC Þf Nt:u: ¼ Nxð1  xB Þg þ Ngð1  xÞxB ð1  xB Þg1 þ MfxC ð1  xC Þf 1 Nc:u: ¼ NgxxB ð1  xB Þg1 þ NCg2 ð1  xÞðxB Þ2 ð1  xB Þg2 þ MCf2 ðxC Þ2 ð1  xC Þf 2 Nb:u: ¼ NxCg2 ð1  xÞðxB Þ2 þ N

DCv ¼

kB b

2

h

g P i¼3

Cgi ðxB Þi ð1  xB Þgi þ M

f P

Cf ðxC Þj ð1  xC Þf j j

j¼3

 i gNKC ðxB εB Þ2

NKB KC 1  frðxB Þ2  gðxC Þ2    x x ðε  εC Þ 2 þ fMKB ðxC εC Þ2 þ frg B C B 1x

(35)

j

(34)

The expression shows that the specific heat Cv is closely related with temperature T and bonding energies of εB and εC. This is because the polymerization is determined by thermodynamic and dynamic conditions of the system. The above results signify that a polymerization system can be dealt with from a physical viewpoint due to the correlation resulted

where the symbols Cgi and Cf are the binomial coefficients, and they take the forms as Cgi ¼ g!=½i!ðg  iÞ! and Cfj ¼ f !=½j!ðf  jÞ!, respectively. In fact, the sum of four types of structural units in the above expressions equals to (N þ M), which is the number of all monomers at the initial reaction stage in the system. As an application, these expressions can be further used to investigate the influences of parameters such as q, r, f and g on the structural parameter of HBPs. To this end, the variations in relative numbers of four types of structural units are presented in Figs. 11e14, where the plots of Nm., Nt.u., Nc.u. and Nb.u. are scaled in (N þ M).

Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

Fig. 11. The plots of Nm., Nt.u., Nc.u. and Nb.u. scaled in (N þ M) against x with f ¼ 4, g ¼ 3, r ¼ 0.05 at different reactivity parameters of q ¼ 0.1 and 10.

695

Fig. 13. The plots of Nm., Nt.u., Nc.u. and Nb.u. scaled in (N þ M) against x with g ¼ 4, r ¼ 0.01, q ¼ 0.01 at different functionalities of f ¼ 3 and 5.

Assembling these results, one can find that with an increase in conversion x, the number of monomers decreases monotonously, the numbers of chain units and branched units increase all the time, while the number of terminal units increases at the initial stage and reaches a maximum, then decreases gradually. These results coincide with the growth mechanism of HBPs. Meanwhile, the influences of parameters q, r, f and g on the change tendency of different structural units are concluded as follows: (i) With an increase in the reactivity parameter q, the number of terminal units increases initially, and then decreases, while the numbers of the other three types of molecular structural units have an opposite tendency as shown in Fig. 11. (ii) With an increase in the stoichiometric parameter r, except the number of monomers always tends to increase monotonously, the numbers of the other three types of molecular structural units all decrease as shown in Fig. 12. (iii) With an increase in the degree of functionality f, the number of monomer units increases, the numbers of chain units and branched units both decrease, while the number of terminal units firstly decreases then increases as shown in Fig. 13. (iv) With an increase in the degree of functionality g, the number of chain unit decreases, the numbers of monomers and branched units both increase, while the number of terminal units firstly decreases then increases as shown in Fig. 14. It should be noted that, upon plotting and calculating, f has been

taken many different values. However, obvious distinction between the same molecular structural units still cannot be observed. The same phenomenon also can be observed in Figs. 7 and 8, which showcase the influences of f on the MSRG. In general, to characterize branched structures of polymers, two physical quantities of DB and FB (fraction of branched-points) are usually considered [16,17,40,50]. They are both closely related to the number of branched units. Here, FB is defined as FB ¼ Nb:u: =ðN þ MÞ, and its plots versus conversion x have been presented as the relative number of branched units Nb.u. in Figs. 11e14. These plots show that FB increases monotonously with an increase in conversion x. But at a given conversion, the tendency of FB is affected by many factors such as reactivity parameter q, stoichiometric r and degrees of functionality f and g. Firstly, a large q leads to an inversion of FB, namely, with the increase in q, FB decreases at the initial reaction stage and then increases at the last stage of polymerization as shown in Fig.11. This is because a large q means that the free groups A prefer to react with groups B rather than groups C, it makes the probability of Cf core monomers turning into branched units become less. As a result, the number of branched units decreases with an increase in q at the initial stage of reactions; Secondly, FB decreases monotonously as either f or r increases as shown in Figs. 12 and 13. This means that the

Fig. 12. The plots of Nm., Nt.u., Nc.u. and Nb.u. scaled in (N þ M) against x with f ¼ 3, g ¼ 2, q ¼ 10 at different stoichiometric parameters of r ¼ 0.01 and 0.1.

Fig. 14. The plots of Nm., Nt.u., Nc.u. and Nb.u. scaled in (N þ M) against x with f ¼ 3, q ¼ 10, r ¼ 0.01 at different functionalities of g ¼ 2 and 4.

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Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

presence of core monomers would lead the number of branched units to decrease. This is due to a large r or f always means that there exist more functional groups C in the system, and as a result, more groups A will be consumed by Cf cores (a Cf core can only becomes one branched unit) so that the fraction of ABg monomers forming branched units becomes less; Thirdly, a large functionality g leads to an increase of FB as shown in Fig. 14. This signifies that a large functionality g is more favorable in the generation of branched units. From the above analysis, we may find that there exists a competition between groups of B and C, which has great influences on the molecular structure and is mainly determined by two factors. One is the reactivity parameter q, the other is the relative number of two functional groups of B and C. According to these results, one can design suitable reactivity parameter q, stoichiometric r and functionalities of f and g to obtain the desired HBPs. 6. Conclusion In summary, an unequal reactivity system consisting of ABg monomers and Cf cores is studied by the method of statistical mechanics, and the explicit expressions of the equilibrium free energy, law of mass action and ESDFs for HBPs with and without a core have been obtained. Then as an application, some interesting thermodynamic properties such as the isothermal compressibility and specific heat have been calculated in terms of the equilibrium free energy and meanwhile the molecular structures in the system have been discussed as well. In particular, to study the effect of unequal reactivity, a reactivity parameter q is introduced, and the MSRG is stressed as a focus to investigate the mean dimension of HBPs. To this end, the recursion formula satisfied by the k-th and (k þ 1)-th MSRGs is derived and the second MSRG is explicitly expressed as a function of the reactivity parameter, stoichiometric parameter and functionality. The results reveal that there are two types of competitions in the system. One is between functional groups of B and C, the other is in the growths of two types of HBPs. In general, the former mainly takes place at the initial reaction stage due to the unequal reactivity of B and C in their consumptions of functional groups A, and the latter mainly occurs in the end of polymerization, at which the number of free groups A is relative insufficient. In the process of competitions, a large reactivity parameter q or functionality g is in favor of the growth of HBPs without a core, while a large stoichiometric parameter r or functionality f is of benefit to the growth of HBPs with a core. According to the above analysis, to synthesize desired resultant HBPs, suitable parameters such as q, r, f and g can be designed to control the competitions. So far, some average polymeric quantities of the polymerization system are obtained by the method of statistical mechanics, and they are consistent with the previous results. This means that the present method is an available tool to treat such reaction systems, but it should be noted that the present study is merely a crude model toward the desired aim. This is because the results obtained in this paper are based on the ideal polymerization. In fact, the practical reactions are so complicated that some factors such as the intramolecular cyclization and steric hindered effect should be considered. Hence as an application of the present method, more effort is required to consider the influences of these factors on the structure and properties of resultant product, as will be performed in the future. Acknowledgment This work is supported by the NNSF of P.R. China under Grant Nos. 21274056, 21274037, 21374028 and the doctoral funds of Langfang Teachers University under Grant No. LSBS201308.

Appendix A. Derivation of the second MSRG For easy of presentation, the second MSRG has been given out straightforwardly by Eq. (30) in the Section 4 of this paper. In order to understand it more clearly, the detailed derivation of the second MSRG is shown as below. As mentioned in Section 4, the second MSRG is closely related to all the possible ways of splitting an i-mer into two moieties denoted 0 0 0 0 by D(i, i ) and Dc(i, i ). In essence, D(i, i ) and Dc(i, i ) can be expressed as functions of the ESDF, respectively, and can be further used to calculate the k-th MSRG for utilizing the polymeric moments. 0 0 According to the physical meanings of D(i, i ) and Dc(i, i ), one can find that for HBPs without loops, they satisfy the restrictions

P i0 P

Dði; i0 Þ ¼ i  1 (A1)

Dc ði; i0 Þ ¼ i  1

i0

When the kinetic differential equation and ESDF are given, Dc(i, 0 0 i ) and D(i, i ) can be expressed as explicit functions of the ESDFs. Combining Eq. (A1) with the kinetic differential equation given by 0 Zhou and Yan [36], the corresponding expressions of Dc(i, i ) and D(i, 0 i ) can be obtained as

Dði; i0 Þ ¼ KB S1 Dc ði; i0 Þ ¼ KB S2 þ KC S3

(A2)

where KB and KC are the equilibrium constants in Eq. (5) expressed by the conversions of x, xB and xC. The detailed expressions of the quantities S1, S2 and S3 take the forms as

S1 ¼ ½iðg  1Þ þ 2Pi0

Pii0 2Pi

S2 ¼ ½ðg  1Þði0  1Þ þ mPic0 ;m c S3 ¼ ðf  m þ 1ÞPi’;m1

Pii0 c Pi;m

(A3)

Pii’ c Pi;m

Substituting Eq. (A2) into Eq. (29), hR2i i and hR2i;m ic can be expressed as

D D

R2i

E

R2i;m

¼ Ec

P b2 KB i0 ði  i0 ÞS1 i2 i0 ¼

b2 P 0 i ði  i0 ÞðKB S2 þ KC S3 Þ i2 i0

(A4)

Then combining Eq. (A4) with Eq. (28) and let k ¼ 2, the Eq. (30) in Section 4 can be obtained. It should be noted that in our calculation, the polymeric moP ments M1, M1c , M2, M2c and the quantity of imP c are involved. P i c i;m The detailed expression of the quantity imPi;m can be evaluated by the k-th moment of HBPs with a core iand it takes the form as

X i

  1  xC þ fxC c imPi;m ¼ fMxC 1 þ 1  gxB

(A5)

A list of all symbols and their meanings

Symbols a monomer with groups A and B, where g
2 denotes the ABg functionality of group B Cf a core monomer with group C and functionality f
3 M, N the numbers of Cf cores and ABg monomers

Z.-F. Zhao et al. / Polymer 55 (2014) 686e697

Q(L), F(L) the partition function and free energy of the system with L bonds P, V, T pressure, volume and temperature εB(εC), vB(vC) the bond energy and bonding volume of TAB and TAC type bonds KB(T), KC(T) the relative bonding probabilities of TAB and TAC type bonds l, lB, lC the equilibrium values of L, LB and LC, respectively x, xB, xC the conversions of groups A, B and C, respectively r a stoichiometric parameter introduced as r ¼ M/N c , uc Pi;m i;m the number of i-mers with a core consisting m reacted C groups and its degeneracy Pi, ui the number of i-mers without a core and its degeneracy Mkc , Mk, Mks the k-th polymeric moments of two types of HBPs and the whole system DPn, DPw the number- and the weight-average DP of HBPs without a core c the number- and weight-average DP of HBPs with a core DPnc , DPw s the number- and weight-average DP of the whole system DPns , DPw PI, PIc, PIs the polydispersities of two types of HBPs and the whole system NB, NC the numbers of unreacted groups B and C in the present system kAB, kAC the reaction rate constants of groups B with A and groups C with A q a reactivity parameter introduced as q ¼ kAB =kAC c hR2 ik , hR2 ik the k-th MSRGs of polymers without a core and those with a core hR2i i, Dði; i0 Þ the MSRG of an i-mer without a core and its decomposition ways hR2i;m ic , Dc ði; i0 Þ the MSRG of an i-mer with a core and its decomposition ways cT, Cv, DUthe isothermal compressibility, specific heat and the change in internal energy Nm., Nt.u., Nc.u., Nb.u. the numbers of monomers, terminal units, chain units and branched units

References [1] [2] [3] [4]

Flory PJ. J Am Chem Soc 1952;74(11):2718e23. Webster OW, Kim YH. Polym Prepr 1988;29(2):310e1. Kim YH, Webster OW. J Am Chem Soc 1990;112(11):4592e3. Mathias LJ, Carothers TW. J Am Chem Soc 1991;113(10):4043e4.

697

[5] Uhrich KE, Hawker CJ, Fréchet JMJ, Turner SR. Macromolecules 1992;25(18): 4583e7. [6] Kumar A, Ramakrishnan S. Chem Commun 1993;(18):1453e4. [7] Kim YH, Webster OW. Macromolecules 1992;25(21):5561e72. [8] Turner SR, Voit BI, Mourey TH. Macromolecules 1993;26(17):4617e23. [9] Kim YH. J Polym Sci Part A Polym Chem 1998;36(11):1685e98. [10] Gao C, Yan D. Prog Polym Sci 2004;29(3):183e275. [11] Voit B. J Polym Sci Part A Polym Chem 2005;43(13):2679e99. [12] Zhou F, Zheng Z, Yu B, Liu W, Huck WTS. J Am Chem Soc 2006;128(50): 16253e8. [13] Voit BI, Lederer A. Chem Rev 2009;109(11):5924e73. [14] Xue Z, Finke AD, Moore JS. Macromolecules 2010;43(22):9277e82. [15] Koenen J-M, Jung S, Patra A, Helfer A, Scherf U. Adv Mater 2012;24(5):681e6. [16] Hawker CJ, Lee R, Fréchet JMJ. J Am Chem Soc 1991;113(12):4583e8. [17] Beginn U, Drohmann C, Möller M. Macromolecules 1997;30(14):4112e6. [18] Ba XW, Wang HJ, Zhao M, Li MX. Macromolecules 2002;35(8):3306e8. [19] Ba XW, Wang HJ, Zhao M, Li MX. Macromolecules 2002;35(10):4193e7. [20] Cameron C, Fawcett AH, Hetherington CR, Mee RAW, McBride FV. J Chem Phys 1997;108(19):8235.  [21] Dusek K, Somvársky J, Smr cková M, Simonsick Jr W, Wilczek L. Polym Bull 1999;42(4):489e96. [22] Wang L, He X-H. J Polym Sci Part A Polym Chem 2009;47(2):523e33. [23] Wang L, He X-H. J Polym Sci Part B Polym Phys 2010;48(5):610e6. [24] Wang L, Yang X-M, He X-H. Chin J Polym Sci 2013;31(3):371e6. [25] Konkolewicz D, Thorn-Seshold O, Gray-Weale A. J Chem Phys 2008;129(5): 054901. [26] Konkolewicz D, Gray-Weale A, Perrier S. Macromol Theory Simul 2010;19(5): 219e27. [27] Yan D, Zhou Z, Jiang H, Wang G. Macromol Theory Simul 1998;7(1):13e8. [28] Yan D, Zhou Z. Macromolecules 1999;32(3):819e24. [29] Zhou Z, Yan D. Polymer 2000;41(12):4549e58. [30] Zhou Z, Yan D. Polymer 2006;47(4):1473e9. [31] Zhou Z, Jia Z, Yan D. Polymer 2009;50(23):5608e12. [32] Zhou Z, Jia Z, Yan D. Polymer 2010;51(12):2763e8. [33] Ba XW, Han YH, Wang HJ, Tian YL, Wang SJ. Macromolecules 2004;37(9): 3470e4. [34] Hanselmann R, Hölter D, Frey H. Macromolecules 1998;31(12):3790e801. [35] Cheng K-C, Wang L-Y. Macromolecules 2002;35(14):5657e64. [36] Zhou Z, Jia Z, Yan D. Polymer 2012;53:3386e91. [37] Wang H-J, Hong X-Z, Gu F, Ba X-W. Sci China Ser B 2006;49(6):499e506. [38] Wang H-J, Hong X-Z, Ba X-W. Macromolecules 2007;40(15):5593e8. [39] Zhao Z-F, Wang H-J, Ba X-W. J Chem Phys 2009;131(7):074101. [40] Zhao Z-F, Wang H-J, Ba X-W. Polymer 2011;52(3):854e65. [41] Gu F, Wang H-J, Zhao Z-F. Sci China Ser B 2011;54(3):438e45. [42] Stockmayer WH. J Chem Phys 1943;11(2):45e55. [43] Sawada H. Thermodynamics of polymerization. New York: NY: Marcel Dekker; 1976. [44] Burchard W. Adv Polym Sci 1983;48:1. [45] Cameron C, Fawcett AH, Hetherington CR, Mee RAW, McBride FV. Macromolecules 2000;33(17):6551e68. [46] Zimm BH, Stockmayer WH. J Chem Phys 1949;17(12):1301e14. [47] Dobson GR, Gordon M. J Chem Phys 1964;41(8):2389e98. [48] Li Z, Ba X, Sun C, Tang X, Tang A. Macromolecules 1991;24(12):3696e9. [49] Rubinstein M, Colby RH. Polymer physics. Oxford: Oxford University; 2003. [50] Schüll C, Frey H. Polymer 2013;54(21):5443e55.