The strategy shown below is based on the gelation theory by Flory1.
Consider there are two kinds of monomers A and B in the system, one with fa functional groups a and the other with fb functional groups b. With the principle of equal reactivity, we can say an arbitrary a selected from the system have a probability Pa of having reacted. And the assumption that reactions between a and b groups on the same molecule are forbidden (no intracellular reaction) enable us to enumerate the probable shape of a certain molecule with cA A and cB B from a root to all the molecule’s branches.
For the convenience of enumeration only, the cA a groups and cB b groups are considered distinguishable. It is because there is no intracellular reaction that the number of ab pairs in the molecule is cA+cB-1. It is easy to prove that in the molecule there is cA•f-cA-cB+1 unreacted a and cB•fb-cA-cB+1 unreacted b. If we choose a free a as the root of the entire molecule (or a free b, if there is no free a), ab pairs can be classified into two types: from a to b and from b to a. The following picture can help in understanding their difference. For the record, their numbers are cB and cA-1 respectively.
Selecting a random a from the system, it will have a chance of Pa to be bonded by b. This Pa can be directly derived from the reaction degree Pf, which will be described later. Here for each free a, the chance it is on a cA & cB configured molecule equals the probability that the particular sequence of cA-1 b have reacted and the remaining cB•fb-cB-cA+1 b have not, while cB a have reacted and the remaining cA•fa-cA-cB have not (the root not included). This probability
is the same for each configuration with cA A and cB B. Hence the probability that any unreacted a group is on a cA & cB configured molecule of any structural configuration is
where ωcA,cB is the total number of configurations. This probability has the following physical picture:
To get the number of cA & cB molecules, it needs to be multiplied with
which equals 
, NA is the number of A monomers.
In order to evaluate ωcA,cB, all the a and b groups in the molecule has been assumed distinguishable. First of all, we need to select those a and b who form the ab or ba pairs. Considering each monomer should has at least one of its functional group bonded, the number of combinations is
The definition of parent is the same as that one of tree, which means the monomer connects a neighbor that is the nearest one to the root. Select cB a from a1 to an shown in the picture. That cB is the number of ab and ba pairs (cA+cB-1) substrates the number of a connected to parents (cA-1), which equals the number of reacted a branchers. The same goes for b. Notice that all the parent-connecting b should be paired with branching a and all the parent connecting a should be paired with branching b. The number of combinations here is
The worry of not that loops may occur in such combinations can be eliminated if the expression given above is understood like this:
In the first step, all the conditions of branching a to parent-connecting b are enumerated, that is cB!. Then, before going to a more complex molecule, we simplify the confirmed structures into many a-bx monomers. This structure is quite similar to Flory’s description in “Monomer Size Distribution Obtained by Condensing A-R-Bf-1Monomers” Chapter IX, Principle of Polymer Chemistry1. Any molecule consists of only a-bx monomers always have one free a. In our model, this free a is the root. Since the root is not an option for any brancher b to bond, it is clear that no loops will form in the next step of enumeration of attaching parent-connecting a to brancher b, which connects those a-bx monomers into one molecule.
Until now, the number all the configurations of distinguished cA A and cB B (their functional groups also distinguished) condensing into a molecule has been derived:
The effect of distinguishing A and B should be compensated by dividing cA!•cB!, while the fact that when counting each functional groups on for example an A monomer, the binomial distribution itself requires the first discussed functional group be different from the second one, makes it reasonable to distinguish functional groups in a monomer. Finally, the total number of cA & cB configurations starts from a root a is
The number of cA & cB configured molecule is
Because PafANA equals the number of total ab pairs in the system, thus it equals PbfBNB, the equation given above has symmetric form, which means that for molecules who have no free a, eq. (4) can be derived by using a free b as the root.
Next, from the definition of the reaction degree Pf, Pa and Pb:
The following equations are obtained:
where x is the total number of ab pairs in the system. Pa and Pb can be expressed using Pf. Finally, the following equation shown in the main body is got:
Our Innovation: The SADP model
Under the principle of equal reactivity, the probability sequence described above
represents the each step of enumeration shown on the right of the following picture:
Our model assumes that the second and above reactions on a monomer have a different reaction probability from the first reaction, to wit the Pb•Pb_d. shown on the right of Fig. 6. For convenience, we call this new method SADP model (Second and Above have Different Probability).
There is no change to the definitions of Pf, Pb and Pa, which means the average expect of the reacted functional groups in the system. The same goes for that expect on an arbitrary monomer. The difference is in the numerical distribution of monomers with zero, one, two, three…of its functional groups having reacted. To be specific, the distribution changes from binomial to non-binomial:
Though the actual numerical distribution without equal reactivity should not be limited to, or just isn’t in the form given in Fig. 7, such inaccurate distribution makes it possible for us to uniform the magnitude of the second, third, fourth…reaction probabilities, to wit Pb_d•Pb. This uniformity is preferred for the mathematical convenience, since other forms of reaction probability has been found quite insufferable when trying to enumerate all the cases.
From Fig. 6, the probability sequence is
That is the case when the change (correction) only happen to b groups. In the previous proof of (7), a free a was used as the root. Under that condition, when considering the influence that Pa_d (a’s correction) would have on the probability sequence, it would involve an annoying item of the root-monomer’s branchers. After one or two steps of thinking, one will find that the probability sequence is even dependent on the structural configuration. Here we suggest a shortcut which skips this burdensome procedure.
First, the final expression of Number of cA & cB molecule with Pa_d=1 is certainly
Deem Pa, Pb, Pa_d and Pb_d as the input parameters of those functions. Because the following part
is independent to Pa, Pb, Pa_d and Pb_d, the formula(a as a root) can be rewritten as:
where 
, Fa(Pa_d,Pa) is the burdensome probability sequence of a functional groups’ enumeration, which is unknown. When Pa_d=1,
. This item of a, though may be related to the configuration, should be irrelevant to Pb and Pb_d. Because that all the bonding actions behind b’s probabilities in the sequence are independent incidents to a functional groups’ bonding actions, we can separate the variables as shown in (9).
Because of the symmetry between a and b, the conclusions from (8) to (9) also work if we choose a random free b as the root. Thus we have (10) and (11).
Since the value of the left side of (11) should be equal to that of (9), the following relationship always exist:
Considering that Pa, Pb, Pa_d and Pb_dare independent to each other, the relationships in (13) and (14) are essential for (12).
Because
References:
[1] “Monomer Size Distribution Obtained by Condensing A-R-Bf-1 Monomers”. Chapter IX. Flory, Paul J. Principle of Polymer Chemistry. 1953.
