EXACT WEIGHT FRACTION DISTRIBUTION I N LINEAR CONDENSATION POLYMERIZATION HANS E. GRETHLEIN Thayer School of Engineering, Dartmouth College, Hanover, N . H . 08766 Since the so-called weight fraction distribution derived by Flory is approximate in the general case, this paper extends the Flory equations to give the true weight fraction distribution over the full range of molecular weight and polymerization parameters for linear condensation polymerization. Equations for true weight fraction are derived from the definition W, = M , N , / ~ M , N , . Expressions for the x-mer molecules weight M , are derived as a function of x for case I without and for case II with a product of elimination during the condensation. For case I the so-called weight fraction is equivalent to the true weight fraction only for type i condensation or for type ii when r = 1. For case II it is never equivalent to the true weight fraction. Since the largest error occurs for small x-mers, the true weight fraction distribution is especially necessary for problems with a small average degree of polymerization S,,.
THE nature of the distribution of the polymer species in linear condensation polymerization was treated by Flory (1936) in a classical paper from a probabilistic point of view. The only major assumption used in that development was that all the polymer species have the same chemical reactivity as the monomer. When six or more carbon atoms separate the bifunctional ends of the linear monomer, this assumption has been found to be valid (Flory, 1953). As a result the set of equations derived by Flory giving the distributions for the number of moles and the weight fractions have been used extensively in the literature by many authors. While the number of mole distributions are exact in the context of the equal reactivity assumption, the so-called weight fraction equations are generally only approximate because of certain simplifications made in the calculation of the molecular weights. When the application involves a high degree of polymerization and a negligible loss in weight due to a product of elimination, this simplification introduces negligible error. Since a rigorous derivation of the weight fraction distribution has not appeared in the literature, this paper extends the Flory equations for weight fraction distribution over the full range of polymerization and molecular weight parameters. The present development deals first with the case involving no product of elimination during the condensation, and then with the case involving a product of elimination. An example of the first case would be the polymerization of a diisocyanate with a diol or a diamine, forming a polyurethan or a polyurea. An example of the second would be the polymerization of a dibasic acid and a diol, giving a polyester and water as the product of elimination.
densation can be obtained as limiting cases of type ii, only type ii is considered here. During a type ii condensation the various lengths of polymers generally referred to as z-mers fall into three classes which require separate treatment in the equation development: 1. The even-numbered 2-mers, terminated with an A and a B group. 2. The odd-numbered z-mers, terminated with A groups. 3. The odd-numbered z-mers, terminated with B groups.
The numbers of x-mers for each class are given by the following equations, which were developed from the equal reactivity assumptions (Flory, 1936). N,(even) = NO'p"1r(t-2)'2(1
Type i. A polymerization initiated by the reaction of a single monomer species generally represented as A-B, where A and B are complementary functional groups. Type ii. A polymerization initiated by the reaction between two different monomer species, generally represented as A - A and B-B. Since equations expressing the distribution of type i con-
t
( 1)
for odd z-mers, A-terminated
(2)
N,(odd B ) = $No/pp-1r(H)/2(1- r p ) 2 for odd z-mers, B-terminated
FUNDAMENTALS
(3)
The total number of molecules is given by summing Equations l , 2, and 3 over x as N=$NQ'[
r - 2rp
+ 1]
(4 1
The weight fractions for the 2-mers are developed with the general format of Equation 5 xN, No
II, = -
Two general types of condensation polymerlabion are possible (Flory, 1936) :
I&EC
for even x-mers N , (odd A ) = *N{p*1r(2-1)'2 (1 - P I 2
Previous Work
206
- p ) (1 - r p )
(5 1
where
II, =
number of segments in all molecules z segments long total number of segments
Thus the so-called weight fraction is actually defined as a segment fraction. Since N O = N i ( 1 -I- r ) / r , using Equations l , 2, and 3, Flory derives the so-called weight fraction or segment fraction for the three classes of x-mers from
&(odd A) = ~ p ~ ~ ~ ( ’ + l ) ’~~)( ~ l / ( T1)
(7)
True Weight Fraction Distribution. Following the format of Equation 9 the weight fractions for the three classes of 2-mers are given by combining Equation 14 with Equations 1and 10, 2 and 11, and 3 and 12, respectively.
&(odd B ) =
(8)
TI’, (even) =
Equation 5 as &(even) = 2~p*’rZ’~(1 - p ) (1 - r p ) / (1
+
T)
+ ~ p * ’ r ( ~ ‘ ) / ~( l~ p ) ~ / + (1
T)
(6)
The main parameters in these equations are r, the initial ratio of A to B, and p , the extent of polymerization. When the polymerization is complete with p = 1 only the odd 2-mers terminated with B groups are present. As a result the number distribution and the so-called weight fraction distribution are given by Equations 3 and 8, respectively. Case I. Exact Weight Fraction Distribution with N o Elimination Product
Under the conditions of equal molecular weight for all segments as in the type i condensation and with no elimination product formed during the condensation, the segment fraction is in fact the same as the weight fraction. As deviations from these restrictions occur, a general way of calculating weight fraction is required. Since the exact weight fraction is defined as m
SV, = M,NJ
M,Nx
2-1
where W , =
,w.. .
(9 )
M , (even) = +M{z
M,(odd
for odd 2-mers, A-terminated
(10)
(11)
+ *M~’(z+ 1) + a ) + (a - l)] for odd 2-mers, B-terminated
(12)
where a = M { I / M { . The denominator of Equation 9 can now be obtained by combining Equations 1, 2, and 3 with 10, 11, and 12 as shown below. m
m
x-1,2,3
...
iM,(even)N,(even) f
M,N, =
z-2,4,6...
5
M,(odd A)N,(odd A )
2-1,3,5...
+
5 .. M , (odd B)N,(odd B )
(13)
t-1,3,6.
An expression in closed form is obtained by carrying out the indicated summation on Equation 13. m
~=1,2,1..
.
+
M a , = $iVo’No’(a r ) / r
+ a ) + (1 - ~ ) ] p ” - ~ r ( ” ~-~ )~/ ~) (~l / +( ar )
(16)
W,(odd A ) = )[z(l
II’,(odd B ) =
+
$[z(l -I- a ) f (a - l)]pp1~(z-1)’2(l - ~ p ) ~ / ( ar ) (17) The weight fraction distributions have the usual parameters, p and r , and the new parameter, a. Since Equations 15, 16, and 17 are weight fractions, the summation over all species must be unity. In fact it can be shown that m
W,(even) z-2,4,6...
+ 5 . II‘,(odd A) + z=1,3,6,.
m
2=1,3,5.
..
TV,(odd B ) = 1.0
When the reaction is complete Equations 15 and 16 become zero as required. Equation 17 applies to the odd 2-mers which are B-terminated and becomes
+ a) + ( a - l)]r(*l)‘z(l
+
-T)~/(u
T)
Identity of Weight and Segment Fractions. Now we can compare the true weight fractions with the so-called weight fractions or segment fractions. When the molecular weight of the monomer A-A is equal to that of B-B (corresponding to a = l ) , the weight fractions (Equations 15, 16, and 17) become the same as the respective segment fractions (Equations 6,7, and 8). When the monomer ratio r = 1, Equation 15, as well as the sum of Equations 16 and 17 becomes
W, = ~ ( 1
M,(odd B ) = $M{(z- 1) = *M{[2(1
(15)
for p = 1 (18)
we can proceed to develop appropriate equations for the weight fractions of 2-mers for any class. Definition of Molecular Weights for x-mers. Assuming that there is no product of elimination during the condensation, the molecular weight of any 2-mer can be defined by the following equations:
+ $M/z = $M{Z(l + a) for even 2-mers A ) = $ i M o ’ ( ~+ 1) + *MO”(Z- 1) = +Mo’[2:(1 + a ) + (1 - a)]
T )
W,(odd) = $[z(l
weight of molecules z segments long total weight
(1 - p ) (1- r p ) (1 4-a ) / (a -k
for all
2,
r=1
(19)
Since Equation 19 does not depend on parameter a, the weight fractions are identical with the corresponding segment fractions in this special case for r = 1, even though the molecular weights of A-A and B-B are not the same. In fact Equation 19 is the same as Equation 2 given by Flory (1936), which applies t o type i or type ii polymerization for equal moles of monomers A-A and B-B. However, the specific distributions for the odd 2-mers, the corresponding weight fractions for the A-terminated and B-terminated 2-mers, are dependent on parameter a, as seen in Equations 16 and 17. Thus, while the total weight fraction for a given odd value of 2 and the segment fraction are the same, the individual weight fractions for the A-terminated and the B-terminated z-mers are not equal to the corresponding segment fraction. Number-Average Molecular Weight and Average Degree of Polymerization. At first glance it may appear because of the new definition of the 2-mer molecular weights given by Equations 10, 11, and 12 that the number-average molecular weight, -Ifn, may differ from that defined by Flory. For type ii polymerization Flory (1936) derived the numberaverage molecular weight as
(14) VOL.
0
NO.
2
MAY
1969
207
and where M O is a weighted average molecular weight defined for type ii polymerization as
+
M{Nd ;Mo”No” No’ No” Starting with the definition of M n as
+
Mo =
bl, =
g
&N,/lV
Table 1. Comparison of Segment Fraction with Weight Fraction and x-Mer Molecular Weight with xMo for a Polyurethane
x-Mer, X
(22 )
x-1,2,3
and recalling that the numerator is given by Equation 14 and the denominator by Equation 4,X n becomes (23)
1 3 5 7 9 11
Segment Fraction,
n*
0.167 0.250 0.208 0.146 0.0936 0 0574
Weight Fraction, WZ
Ratio, RT
0.0543 0.250 0.237 0.174 0.115 0 0714
3.08 1.0 0.88 0.84 0.815 0.804
-
By solving Equation 24 for Mo’ and substituting into Equation 23, the number-average molecular weight in this paper becomes the same as that defined by Equation 20. Because the number-average molecular weight does not depend on parameter a and agrees with Florg’s definition, it can also be related to the average degree of polymerization by
Mn =
(25 )
XMO
250
2,300 f/8 4,350 1/16 6,400 1/32 8,450 1/64 10,500
-
Moles 1
_
Totals
2
3
767
2300 3835 5370 6900 8440
~
Wt., G‘. 1800 500
__
2300
Monomer ratio, T = 3 Mol. wt. ratio, a = __ 250 = 0.139 1800 Extent of polymerization, p = 1,O M o = -____ 1!1800) 2i250) = 767 3
+
Average degree of polymerization, %, = FT l+
where
t
=3
2
f n
=
= -No N
r-2pr+1
Comparison of Segment Fraction with True Weight Fraction. The so-called weight fraction or segment fraction given in Equation 5 can be defined with the aid of Equations 25 and 26 as
TI,
=
X11f&,
-
MnN and the true weight fraction can be expressed as
of x as shown jn Equation 30 and so is constant for every even 2-mer. For an odd x the error varies with 2 as shown in Equations 31 and 32. ;is x becomes large, the error approaches a fixed value given by Equation 30. Naturally when a = 1, R, becomes unity for all 2. Usually the interest in weight fraction distribution is for complete polymerization with p = 1. In the general case with r # 1, only Equation 32 is of interest. When 2 = 3n = (1 f r)/(l - r ) , Equation 32 becomes equal to the unity, which is also obvious from Equations 25 and 29. fnM0
Thus the ratio of the segment fraction to the weight fraction for a given value of x, which is a measure of the discrepancy between the two concepts, is XMO !If,
(29 1
R, = -
By referring to Equation 24 for .UOand Equations 10, 11, and 12, respectively, for X,,ratios can be defined for the three classes of 2-mer8 as R,(even) =
+r) (1 + r ) (1 + 2(a
a)
With Equations 30, 31, and 32 it is now possible t o determine quantitatively to what extent the simpler treatment of segment fraction as a weight fraction is reasonable for any values of the parameters. First, we note that the error does not depend on the extent of polymerization, p, for a given value of a and r . The error for an even x is not a function 208
lbfz
0.748
m
Polymerization parameters Monomer Mol. Wt. A - A = diol 1800 B-B = diisocyanate 250
Equation 21 can be expressed as
N,
2
l&EC
FUNDAMENTALS
R,,=--
-1
*Ifn Thus the weight fraction and the segment fraction are always equal for 2: = 3n. When a < 1, R, is greater than 1 for x < f n and R, is less than 1 for z > 3n. JT’hen a > 1, R, is less than 1 for 5 < 3n and R, is greater than 1 for z > Bn. In Table I an example is given of the formation of a polyurethane from a diol and a diisocyaiiate. Since an average degree of polymerization of 3 is desired, a monomer ratio of r = 3 is used. The largest discrepancy between the segment and weight fraction is for the unreacted diisocyanate monomer Tvith R1 = 3.08. For the trimer R3 = 1.0 and for higher x-mers Rx approaches 0.748. Also shown in the table are the 2-mer molecular weight, JI,, and zJf0, which is an approxiniate z-mer molecular weight implicit in the so-called weight fraction or segment fraction. Comparing the last two columns in the table indicates that the x-mer molecular weight does not increase proportionately to 2, since in general the alternating segments are not of the same molecular weight. In fact it n’as this example which initially brought to the author’s attention the large difference in the calculated amount of the unreacted monomer using I& (0.167 X 2300 = 358 grams) as compared t o that calculated from the number of moles of monomer (3 X 250 = 125 grams).
Case II. Exact Weight Fraction Distribution with Elimination Product
Definition of Molecular Weights for x-mers. When a condensation polymerization such as between a diol and a dibasic acid involves a product of elimination, a further modification has to be introduced for calculating the molecular weight of an z-mer. The problem can be visualized by considering t'he formation of a dimer shown below.
0
0
I/
--: /I
HO-C-R'-C-OH
+ HO-R"-OH
+
7 -
Jf
':,1 0
0 'I
'I
H-0-C-R'-C - 0-R"-OH - . - - w - mol nid' L
rvz = [z(I + a ) +
+ H2O Me
,
[(I
A monomer molecular weight can be thought of as a fixed segment weight (md or m d ' ) which propagates into the polymer and a terminal part, AI,, which is equal to the molecular weight of a product of elimination. Thus
+ Me
and N {' = md'
+ Jf,
(33)
By defining the monomer segment weight ratio a s a = m:'/mo' which is analogous t'o a in Case I, the exact molecular weights for the three classes of z-mers are given by
+ a ) + Me for even z-mers Al,(odd A ) = $mo'[z(1 + a ) + (1 - a ) ] + 31, M,(even) = +m(x(I
for odd z-mers, A-terminated Xz(odd B ) = $mi[z(l
+ a ) + (a - l)] +
(35)
(36)
True Weight Fraction Distribution. In deriving equations for the weight fractions of z-mers the definition given in Equation 9 is used. ;in expression for the denominator of Equation 9 for Case I1 can be derived carrying out' the summation shoivn in Equation 13 by using Equations I , 2, and 3 with 34, 35, and 36 to give
(37 1 Following the format of Equation 9 the weight fractions for three classes of z-mers are given by combining Equation 37 with Equations 1 and 34, 2 and 35, and 3 and 36, respectively. TT',(even) =
[z(l
(a
+ r ) + (-Jfe/mo') (r - 2pr + 1
for odd z-mers A-terminated
+[z(I + a )
for all z-mers, r = 1, a = 1 (42) which shows that the weight fraction still depends on the parameter M e / m { . Thus only when there is n o elimination product or when -11, is very much smaller than vi; can the segment fraction and the weight fraction be considered equivalent concepts. Number-Average Molecular Weight. V'ith the definition of the molecular weight of the z-niers given by Equations 34, 35, and 36 the definition of the number-average molecular weight given by Equation 22 becomes
If an attempt is made to relate the average degree of polymerization to the number-average molecular weight corresponding to Equation 25, a modification must be made so that M n = B,Jlo JI, (42 1 where
+
Mo=
(39)
+ (a - 1) + ( 2 * ~ e / m o ' ) ] p " - 1 r ( ~ 1-) ' rp)2 2(1
(a
+r) +
+
(Jfe/mo')(T - 2pr 1) for odd z-mers, B-terminated
+r) (1 +
mo'@
(43)
and Bn is the same as defined in Equation 26.
+ a ) + (I - a ) + (2~lie/mol)]pz-ir(r+i)'2 (1- p)' (a + r ) + (nf,/mo')(r - 2pr + 1)
B) =
As in Case I when r = 1 type ii condensation can be treated as type i in that one equation (41) gives t'he weight fractions for all z-mers, which is analogous to Equation 19. However, in Case I1 the interpretation of Equation 41, which depends on (Y and Me/md does not coincide with a segment fraction as in Case I for Equation 19. In fact, even for the restricted interest when a = 1 corresponding to a pure type i condensation Equation 41 becomes
(38)
W"(odd A ) =
",(odd
for all z-mers, r = 1 (41)
+ a ) + (2.1~e/m")]p"-1r"2(1 - p ) ( l - r p ) for even z-mers
$[z(I
+a)+
(34)
JIe
for odd z-mers, B-terminated
@Jfe/m0')1~**(1 - P)' (2Jfe/mo') (1 - P)]
(32)
,If e
= m i
ilgain, it can be shown that the summation of Equations 38, 39, and 40 over all z gives unity as required. These weight fraction distributions are dependent on the usual parameters, r and p and a , the ratio of the monomer segment weights, and Me/mo', the ratio of the molecular weight of the elimination product to the segment weight of monomer A-A. Equations 38, 39, and 40 are the most general weight fraction distributions for linear coiidensat'ion polymerization and are valid for the full range of the parameters r, p , a, and Me/mo'. Comparison with Case I. For the special case where r = 1 Equation 38 as well as the sum of Equations 39 and 40 becomes
(40)
Nomenclature
A = function group A-A = linear monomer terminated with A functional groups n = ratio of monomer molecular weights, Jid'/Mo' B = complementary functional group t o A B-B = linear monomer terminated with B functional groups M e = molecular weight of product of elimination during condensation polymerization -Ifn = number average molecular weight M, = molecular weight of an z-mer VOL.
8
NO.
2
MAY
1969
209
= weighted average molecular weight of a segment
iM0
Mi
= molecular weight of A-A Md’ = molecular weight of B-B m,,’ = fixed part of molecular weight of
= ratio of A-A to B-B = N,,’/N,,‘I W , = true weight fraction for 2-mer
T
z
A-A appearing in
2-mers
Z a
m,,” = fixed part of molecular weight of R-B appearing in 2-mers N = total number of molecules No = initial number of molecules = (No’ N i l ) N{ = initial number of -4groups No” = initial number of B groups N , = number of molecules of 2-mer p = extent of polymerization R, = ratio of segment fraction or so-called weight fraction to true weight fraction for 2-mer
a
+
11,
= number of segments in z-mer = average degree of polymerization = ratio of fixed parts of monomer molecular weights, m{’/m{ = so-called weight fraction or segment fraction
Literature Cited
Flory, P. J., J. Am. Chem. SOC.68, 1877 (1936). Flory, P. J., “Principles of Polymer Chemistry,” p. 71, Cornel1 University Press, Ithaca, N. Y., 1953. RECEIVED for review December 27, 1968 ACCEPTEDFebruary 24, 1969
MEASUREMENT OF EFFECTS OF MIXING OF CHAIN CENTERS I N A NONUNIFORMLY INITIATEDP H O T O P O L YMERIZATION L O W E L L Y E M I N I A N D F R A N K B. H I L L Brookhaven National Laboratory, Upton, N . Y . 11973
Apparatus has been developed for the study of effects of mixing of chain centers in nonuniformly initiated photoreuctions. The experimental method is based on the use of a quartz reaction vessel, partially illuminated with ultraviolet light, equipped with a stirrer, and outfitted otherwise as a dilatometer. In preliminary experiments involving the direct photoinitiated bulk polymerization of methyl methacrylate, the rate of polymerization varied with stirring speed, effective optical thickness, and incident intensity approximately in accordance with theoretical predictions. Evidence was found for the presence of spurious initiators, and for a hysteresis effect, resembling the result of the compounding of effects of viscosity and mixing. Methods of overcoming these difficulties were suggested.
MIXING phenomena are important in determining the ob-
served reaction rate and product distribution in optically dense photoreaction mixtures. The accurate prediction of mixing effects requires a detailed knowledge of the mechanism of the reaction of interest, the optical properties of the system, the physics of the radiation used, and the mixing processes themselves, Seldom will adequate information be available for a given system to permit prediction out of hand of the observed rate, Usually experimental characterization will be required. I n addition to serving the purpose of characterization, experimental studies are useful in the first place in understanding the nature of the interaction of chemical kinetics and mixing processes in the presence of nonuniform initiation. I n the present paper apparatus is described for the study of effects of mixing of chain centers in photoreaction mixtures which have artificially been made strongly absorbing, and data are presented for the direct photopolymerization of methyl methacrylate. Method and Theory
I n optically dense systems mixing effects may be distinguished which result from nonuniform depletion of reactant 1
210
Present address, Gamma Process Co., Morton Grove, Ill. l&EC
FUNDAMENTALS
(Hill, Reiss, and Shendalman, 1968), or from nonuniform production of reactive intermediates (Hill and Felder, 1965; Hill and Reiss, 1968). The measurement of such mixing effects requires apparatus with a mixing time scale which may be varied around the average lifetime of reactant or intermediate, In the present paper, the apparatus described has a mixing time scale on the order of seconds, comparable with attainable lifetimes of chain centers in vinyl polymerizations, and it is the mixing of such centers that was studied experimentally. Reactant mixing experiments with the present apparatus would be confined to the perfect mixing region, since reactant lifetimes are commonly on the order of hours. The radiation used in the present study was ultraviolet light. Though mixing effects may be found with other radiations, such as gamma-rays and beta-particles, the study of mixing effects requires an accurate knowledge of the spatial distribution of rates of energy absorption. This requirement is most readily met at present with ultraviolet light. A schematic diagram of the reaction vessel used is shown in Figure 1.
A volume V L is illuminated with a parallel, effectively monochromatic, uniform beam of weakly absorbed radiation. For these conditions, and with negligible reflection, the absorbed intensity is uniform everywhere within the lighted volume with the value pIo, where fi is the linear absorption
