The conversion
dependence of the average mean square radius of gyration for the selfcondensing vinyl
polymerization
Zhang Shuwen, Ba Xinwu, Han Yinghui,
Wang Sujuan, Wang Haijun
(College of Chemistry & Environmental Science, Hebei University, Baoding 071002,
China)
Abstract By means of kinetic
differential equation approach, the mean square radius of gyration of the hyperbranched
polymers formed by selfcondensing vinyl polymerization was investigated explicitly.
Averaged over the molecular weight distribution, the explicit expressions of the number,
weight and zaverage mean square radii of gyration are given as a function of conversion
of double bonds, in which the contribution of the unreacted monomers in the hyperbranched
polymer systems is included and excluded, respectively. The variation of the average size
of the hyperbranched polymers during the selfcondensing vinyl polymerization can be
predicted to conclude that the weight and the zaverage mean square radii of gyration
increase sharply at the end of reaction, which enlarge over ten times as the conversion of
double bonds rises from 0.99 to 0.999.
Keywords Hyperbrabched polymer, radius of gyration, selfcondensation vinyl polymerization
1. INTRODUCTION
Hyperbranched polymers have received considerable attention in recent year partly because
they do show properties very similar to dendrimers (e.g. high functionality, low
viscosity, and so on), which are of particular interest for the utilization in catalysis,
supermolecular chemistry and biochemistry, ^{[1]} and their syntheses are much
simpler than those of dendrimers.^{ [2,3]} In 1995 Frechet and coworkers reported
a new way of forming hyperbranched polymers based on vinyl monomers and named it as
"selfcondensation vinyl polymerization" (SCVP).
^{[4]} Since then living cationic,^{[4]} nitroxidemediated radical, ^{[5]}
atom transfer radical, ^{[6]} and group transfer polymerizations ^{[7]}
have been applied to the synthesis of hyperbranched polymers. The theoretical description
of SCVP leading to hyperbranched polymers was proposed by Muller and Yan to give the
molecular weight distribution, the branching density and the other characteristic mean
values. ^{[8,9]} As it is well known, the physical properties of polymer materials
depend crucially on molecular parameters, such as the molecular number distribution, the
average molecular weight, the branching density, the mean square endtoend distance and
the mean square radius of gyration and so on. ^{[1216]} The mean square
endtoend distance and the mean square radius of gyration are the two most important
values in conformational statistics of polymer chains. For branched or hyperbranched
polymer systems, only the mean square radius of gyration is meaningful. From the
architecture of a hyperbranched polymer, one can see that the mean square radius of
gyration is smaller than that of a linear polymer chain of the same molecular weight due
to the branching effect. The mean square radius of gyration can be averaged over the
molecular weight distribution to give the average mean square radius of gyration, which
increases as the conversion of functional groups or reaction time, and is experimentally.
For example, the zaverage mean square radius of gyration is generally determined from
light scattering experiments. ^{[16]} Experimental measurement of the structure
functions at small momentum transfer can yield the zaverage mean square radius of
gyration varying with the conversion of groups or reaction time, and therefore
understanding the variation of the average mean square radius of gyration with the
conversion or reaction time may help to improve present knowledge about the architecture
of the elementary structure and formation kinetic. From the theoretical point of view,
there are two methods being used to calculate the average mean square radius of gyration.
One is the computer simulation based on the formation kinetic, which is not discussed in
detail in this paper. ^{[19, 20]} Another is the analytical technique, such as the
statistical method, ^{[13, 14]} stochastic theory of branching processes, ^{[15]}
the kinetic approach method ^{[17,18]} and so on. It should be point out, to our
knowledge, that no analytical technique has been proposed to give the conversion
dependence of the average mean square radius of gyration during the selfcondensing vinyl
polymerization.
In this paper, we focus our attention on the conversion dependence of
the average mean square radius of gyration for hyperbranched polymers systems generated by
selfcondensing vinyl polymerization. We will first give a brief review of the kinetic
differential equation approach and the concentration of nmer P_{n},
^{[8]} and then a decomposition formula for the combinatorial coefficient of the
concentration of nmer will be proposed. The statistical interpretation of this
formula, which is similar to the argument proposed by Dobson, Gordon ^{[15]} and
some of the authors ^{[17,18]} for the branched polymers, is shown in detail to
give the mean square radius of gyration. Finally, we will give the analytical expressions
of the number, weight and zaverage mean square radii of gyration dependence on the
conversion of double bonds, in which the contribution of the unreacted monomers in the
hyperbranched polymer systems is included and excluded, respectively.
2. THE MEAN SQUARE RADIUS OF GYRATION
According to the reaction scheme proposed by Muller and Yan, the kinetic differential
equation and chemical kinetic equation for selfcondensing vinyl polymerization can be
given as ^{[8,9]}
(1)
(2)
where P_{n} is the concentration of nmers, x is the
conversion of double bonds, W_{0} is initial concentration in the reaction
system, and k is the effective rate constant of the reaction. In obtaining above
equation, the assumptions of ideal network polymerization were retained, i.e., all
functional groups of the same type are equally reactive, all groups react independently,
and no intramolecular reactions occur in finite species. With the initial condition , where is Kronecker symbol, the concentrations of nmers can be derived as
^{[9]}
(3)
with
(4)
where C_{n} is the combinatorial coefficient.
Let us now give a decomposition formula of the combinatorial
coefficient C_{n}. Dividing Eq.(1) by Eq.(2), one can obtain
(5)
Substituting the concentration of the nmer P_{n}
given by Eq.(3) into Eq.(5), a decomposition formula for the combinatorial coefficient C_{n}
can be given
(6)
In order to give the statistical interpretation of Eq.(6) more clearly, we may rewrite it
in symmetric form
(7)
with
(8)
For an arbitrary nmer in the ensemble of molecules, there may
be in general (n1) bonds formed during the polymerization process without
consideration of the intramolecular reaction. Considering an arbitrary bond in the nmer,
where the monomers are defined as the mass points for simplicity, there may be two
moieties if this bond is imagined to be cut. The number of mass points found in the two
moieties of molecule produced by cutting this bond is i and (ni)
respectively. From Eqs.(7) and (8), we see that the term n1 on the left side
is the number of bonds in a nmer, and the term D(n.i) on the right
side should be the number of bonds whose splitting produces two moieties of i and (ni)
mass points. The summation in Eq.(8) with respect to i gives the total number of
bonds in the nmer.
Next the mean square radius of gyration for the hyperbranched polymers
is calculated with aid of a decomposition formula given by Eq.(7). The square radius of
gyration for a hyperbranched polymer consisting of n mass points with fixed
conformation is defined as
(9)
where r_{i} is the distance of the ith mass point from the center
of gravity of the molecule. A simple argument, proposed by Zimm and Stockmayer leads to
the transformation ^{14}
(10)
where r_{ij} is the distance from the ith mass point to the jth
one. With cyclic molecules being excluded, there is a unique path of bonds leading from ith
mass point to the jth one in the nmer, in which the number of bonds on the
path can be varied from 1 to (n1). The mean square radius of gyration , which averages the fluctuations in time of due to Brownian motion, can be expressed as
(11)
Using the method proposed by Dobson and Gordon ^{[15]}, the
mean square radius of gyration in
Eq.(14) can be reformulated as
(12)
where b is the effective bond length. It should be noted that the assumption of all
bond lengths in a hyperbranched polymer being equal was taken. The index h in the
summation is used to denote the hth bond in the nmer. The term and on the right hand side of Eq.(12) are the number of mass points
associated with the two moieties which produced by imaginary cutting the hth bond
in the nmer. The summation runs over all the bonds in the nmer. As being
mentioned above, in Eq.(8) is the number of
bonds whose splitting produced moieties of i and (ni) mass points,
respectively. Then the mean square radius of gyration in Eq.(12) can be expressed as
(13)
The summation in Eq.(13) with respect to i means that splitting
the nmer produces two moieties of i and (ni) mass points,
respectively. Since the number of mass points in those two moieties varies from 1 to
(n1), the summation in Eq.(13) contains all the moieties with different
number of units.
3. THE AVERAGE MEAN SQUARE RADII OF
GYRATION
For most majority synthesis of hyperbranched polymer systems, many different isomeric
structures and a broad molecular weight distribution for hyperbranched polymers are
obtained. Therefore only the average statistical parameters, for instance, zaverage mean
square radius, can be measured by light scattering techniques. ^{[16]} In order to
calculate the average mean square radii of gyration, the number, weight and
zdistribution of the hyperbranched polymers was defined as ^{[9]}
(14)
(15)
(16)
where are the polymer moments defined as
, ,
(17)
The first polymer moment is the material
balance condition.
With the aid of the mean square radius of gyration in Eq.(13), the
number, weight and zaverage mean square radii of gyration for hyperbranched polymer
systems can be defined as follows
(18)
(19)
(20)
where is defined as
(21)
Substituting the mean square radius of gyration in Eq.(13) and number distribution N(n)
in Eq.(14) into Eq.(21), and letting k=2, the quantity can be obtained by means of direct summation
(22)
Unfortunately, the two quantities and cannot be evaluated
analytically by means of direct summation. Since the mean square radius of gyration in Eq.(13) is independent of the conversion of
double bonds x, we can choose x as a variable to differentiate both the right and left
sides of Eq.(21). Letting k=1 yields
(23)
Substituting quantity in Eq.(22) into the above equation
yields
(24)
With the boundary condition for x=0, the
quantity is obtained by direct solution of
the above differential equation
(25)
Consecutively, the quantity can be
calculated in the same manner as that of quantity . Since the calculation procedure is straightforward, here we only give the
result as follows
(26)
Using the above results in Eq.(22), (25), (26) and the polymer moments in Eq.(17), the
number, weight and zaverage mean square radii in Eq.(18)(20) can be given
explicitly
(27)
(28)
(29)
The results given by Eq.(27)(29) can be used to predict the variation
of the average size of the hyperbranched polymers during the selfcondensation vinyl
polymerization.
Figure 1. Dependence of the average mean squre radii of gyration on x
¡ö: ¡ñ:¡ø:
Figure 1 shows the numerical calculations of
the average mean square radii of gyration as function of conversion of double bonds, x.
It can be seen that the average mean square radii of gyration increase much more slowly at
the beginning of the reaction, and increase sharply at the end of reaction. The weight
and zaverage mean square radii of gyration increase over ten times as the conversion of
double bonds rise from 0.99 to 0.999. In fact the average mean square radii of gyration
reach infinite when the conversions of double bonds are near the maximum value x=1. The
results also show that the number average mean square radius of gyration increases slowly than the weight and the
zaverage mean square radii of gyration as the conversion of the double bonds increases,
since the former is divergent as the logarithmic function. The hyperbranched polymers can
be prepared by polymerization of AB_{f} type monomers, where A and B
are two different functional groups capable of reacting with each other, and f is
the number of B groups. ^{[2,3]} Here we make the comparison between
zaverage mean square radius of gyration of the hyperbranched polymers obtained from the
selfcondensation vinyl polymerization with that produced in a polycondensation of AB_{2}
type monomers. The zaverage mean square radius of gyration of the hyperbranched polymers
formed by AB_{2} type polycondensation was given as ^{[18]}
(30)
where y is the conversion of the A groups.
Figure 2 shows the numerical calculations of the zaverage mean square
radii of gyration, and , as function of the conversions. It can be seen
that the difference between the two quantities is smaller than 1.0% when the conversions
is over 0.99. As the conversions are near the maximum value, x=1 and y=1, the two
quantities have the same asymptotic behavior.
Figure 2. Dependence of the zaverage mean
square radii of gyration, and on the conversions
¡ñ: ¡ø:
In the experimental procedure, the low molecular weight materials, such
as residual monomer etc., are usually separated from the polymer systems by precipitation.
The number, weight and zdistribution of the hyperbranched polymer systems without the
unreacted monomers take the forms
(31)
(32)
(33)
with
(34)
where , and are the polymer moments
without the unreacted monomers. The number, weight and zaverage mean square radii of
gyration without the unreacted monomers can be evaluated as
(35)
(36)
(37)
With the aid of the number distribution given by Eq.(14) and the polymer moments without
the unreacted monomers given by Eq.(34), the average mean square radii of gyration without
unreacted monomers can be given as
(38)
(39)
(40)
Furthermore, the contribution of the low molecular weight materials, such as dimer etc.,
to the average mean square radii can also be calculated without difficulty, and are
omitted in the present paper.
4. SUMMARY AND CONCLUSIONS
From the chemical kinetic equation given by Eq.(2), and branching density definition which
is important parameter characterized the topological structure of hyperbranched polymers,^{
[810]} the conversion of double bonds, x, can be easily derived as a
function of the reaction time, or the branching density. The conversion of double bonds,
x, as well as reaction time and branching density, can be measured by means of NMR and
FTIR and other techniques. ^{[47]} The zaverage mean square radius of gyration
is generally determined from light scattering experiments. Thus, our results are
convenient to compare with the experimental data since all quantities in the formulas are
experimentally accessible.
The results obtained in the paper were compared with that produced by
the polycondensation of AB_{2} type monomers. The numerical calculations showed
that the difference between the two quantities is smaller than 1.0% when the conversions
is over 0.99, and the same asymptotic behavior were also given as the conversions are near
the maximum value, x=1 and y=1. Although the topological structures of the hyperbranched
polymer and dendrimer are similar, but the branching density of the hyperbranched polymer
is smaller than that of the dendrimer, where the branching density has the value 1.0 for
the latter, so a number of linear units exist inside the hyperbranched polymer skeleton.
This means that the molecular weight, as well as the mean square radius of gyration, of
the hyperbranched polymers increases without limitation as the conversion of function
group increase, and the mean square radius of gyration is larger than that of the
dendrimer of the same molecular weight.
The explicit expressions of the average mean square radii of gyration
obtained in this work are based on the assumptions of the statistical ideal network
polymerization, which is the basic model used to describe the polymerization process, and
may not be fulfilled in reality, ^{[9, 11]} and therefore certain deviations are
to be expected when comparing with experimental results. The average mean square radii of
gyration can be evaluated by means of the computer simulation technique whenever
statistical nonideal network polymerization is taken into consideration.^{[9, 10, 20]}
The analytic results in the paper presented an effective way of understanding the physical
properties of hyperbranched polymers during coagulations, and can be used to examine
whether or not the simulation is correct for a given model.
Acknowledgment. This work was supported by the National Natural Science Foundation
(20174008) and the Foundation for University Key Teacher by the Ministry of Education
(680) of China.
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