MOLECULAR SIZEDISTRIBUTION IN POLYDISPERSE SYSTEMS
May, 1941
reduction of an ion gives rise to phenomena which simulate the behavior of a condenser
[CONTRIBUTION FROM
THE
1215
of very large capacity. AWERST,MASS.
RECEIVED DECEMBER 2, 1940
STERLING CHEMICAL LABORATORY, YALEUNIVERSITY ]
Molecular Size Distributions and Depolymerization Reactions in Polydisperse Systems BY ELLIOTTMONTROLL* When experimenting with long chain polymers i t is desirable to use molecules which are all composed of the same number of monomeric elements. Since such a homogeneous system is difficult to prepare, one must often be satisfied with more or less heterogeneous mixtures. An indication of the molecular size distribution of this type of system can be obtained by applying statistical methods to the various molecular weights that can be determined experimentally. If a long chain molecule in a system of N similar polymers consists of p monomeric elements or fundamental units (for example, the monomeric elements in cellulose are glucose units), i t will be a +-mer and its molecular weight will be denoted by ATfi. Letting np be the fraction of p-mers in the system and m the molecular weight of a monomeric element, Nn, is the total number of p-mers and M, = mp. Now any experimental method (such as chemical analysis, vapor or osmotic pressure measurements) which in effect involves the counting of molecules, measures the “number average’’ molecular weight The average value resulting from a procedure depending on weights of the molecules (e. g., Staudinger’s viscosity method) is given by the “weight average” molecular weight :
can be expressed in terms of the Mi’s
or, since
c9.
=1
M1M2. . .Mi/mi (6) I n a polydisperse system in which the molecular size distribution has a single maximum, this distribution might be almost normal. As has been discussed by Lansing and Kraemer,’ a logarithmically normal distribution function is applicable when there is reason to believe that a strong deviation toward the high molecular weight components exists. Since only two moments are required to calculate normal and logarithmically normal distribution functions, they do not make use of all the possible experimental information. We will now proceed to derive a distribution function which involves all the measurable moments, and then we will develop a theory of depolymerization of polydisperse systems with arbitrary initial distributions. Almost Normal Molecular Size Distributions. -The fraction of p-mers in a system with a normal distribution2 of molecular sizes is bi
=
no(p) = e - ( w - ~ ) * / z ~ z / r & r
(7)
where p1 is the average degree of polymerization or first moment of the distribution and u2 is the Mz = ~ p ~ ~ ~ n , / ~ p ~ ~ (2) f p p pmean deviation, w - p;. Since any continuous m Sedimentation equilibrium measurements in an function, f@), such that f If@) 2dp < a ,can be - m ultracentrifuge make available the “e-average” expanded as a product of a Gaussian error funcmolecular weight’ tion (7) and a linear combination of Hermite polyMa = c p M h J c p M & p (3) nomials, an almost normal distribution function I t is now apparent from (1)) (2) and (3) that the can be written moments of the distribution function, np, the i-th of which is
1
pi =
* Sterling Research
CppinP
I.
(4)
Fellow, Yale University. (1) See Kraemer’s article on polydisperse systems in “The Ultracentrifuge,” edited by Svedberg and Pedersen, Oxford Press (1940) ; or Lansing and Kraemer, THISJ O U R N A L , 57, 1369 (1935)
-”
(2) The treatment for logarithmically normal distributions similar to those of Lansing and Kraemet’ hut involving three observable moments proceeds in a similar manner. See, for example, A Fisher, “The Mathematical Theory of Probabilities,” New York, N. Y., 1926, p. 235.
ELLIOTT~MONTKOLI.
l2ll(i
The first few Hermite polynomials are Ho(2) = 1 HI(z) z H*(z) = 22 H&) = 23
-I
- 32
and all these polynomials satisfy the ortho normalization condition 1
"
.\/2, f
z ~ , ~ ( z ) ~ , , ( z ~ e - ~ *= / ' dv'nG!6,h,,z e
(sa)
- m
V~i1.12:
differentiate experimentally between a molecule of say 300 monomeric elements and one of 301, the error introduced by replacing the sums by integrals is not large. The introduction of the various average molecular weights makes n, = p - i.Mdm - P ) ' / 2-_ ~2 1 - A!. F{9(Md?!2?) + , I. ,
U V G
a*
Mt(M2
I
:3!d
- M~)/NL'
Xa = M,(SM: - 3MIMI
a,,
= 1 if m = n and zero otherwise. ing z = ( p l - p)/u in (9a):
Substitut-
,
\
U
(12)
+ M2M3)/wba
Physically, u! gives a n indication of the sharpness of the distribution in such a way that as D 1,1creases, the distribution curve broadens, while the skewness or deviation from normal is indicated by (Yb) X g SO that as XB +0 the curve approaches a norThus multiplication of both sides of (8) by Hm- mal one. When X3 is negative, molecular sizes 'l - clp and integration from ,b = - to smaller than the most probable ones predominate over those less than the most probable, and vice p = m yields the constants versa for positive n3. In Fig. 1 are plotted three distribution functions for systems with number average molecular weights MI = 330m and weight average molecular weights Mz = 3 . 5 4 ~ . Curve 'To translate the c's into experimental moments (1) is the normal distribution which corresponds and average molecular weights we proceed as to X3 = 0. Curve ( 2 ) results from choosing a zrollows average molecular weight M, = 357m. This m m choice makes X.,/6u3 = -0.359, the negativeness C O = f Honpdp = f n,dp = I of which leads to a predomination of less than "most probable" molecular sizes. Curve (3) shows the long chain predomination that comes from choosing Ma = :339m. Here Xd/Aa3 = f0.480. The skewness is as delicate to small differences in Ma as i t is in Fig. 1 only when the M3's are close to the value which makes the curve normal. Depolymerization of a Mixture of Long Poly-_ 3PW -k e - _ 2Pi __ mers.-A complete investigation of the kinetics U? Using these values for GI, L? and 63 and abbreviating of decomposition of long chain molecules has two aspects: the time variation of the degree of deX: = 2': - :%pip2 p 3 , np becomes polymerization, and the distribution of various chain lengths a t a given time for a given degree of (11) depolymerization and a given initial distribution of chain lengths. The size distribution aspect Although it is possible in principle to determine has recently been analyzed3for a system in which. moments higher than the third from ultracentriI . All initial molecules are of the same molecular fuge analysis the errors would accumulate rapidly, weight. 2. The accessibility to reaction of a so it would be advisable to cut off the expansion with the third Hermite polynomial; however, un- bond is independent of its position in a chain and less a system deviates radically from a normal independent of the length of its parent chain. 3. one, (11) should be quite accurate. Actually the The bonds of all the chains i n the mixture a t any experimental moments pi = MIMs. . M,/m' are given time are equally atccessi1;le to reaction. sums, while the pt's in equation ( 1 1 ) are integrals. The difficulty of preparation 01 uniform samples (3) Montroll and Simha, J . Chew P h y s . , 8, 7'21 ( I ! M l ) ) ReferBut, inasmuch as the number of molecules to be ences to past work. experimentirl and theoreticdl. ;ire y i v c I I in lhi. averaged over is large, and since it is difficult to paper.
(4 +
+
MOLECULAR SIZE DISTRIBUTION IN POLYDISPERSE SYSTEMS
May, 1941
for depolymerization makes it desirable to eliminate the first assumption; so, using the results of the last section we will now investigate the degradation of mixtures with an arbitrary initial distribution and especially those whose initial distribution is almost normal. If an initial material is homogeneous the depolymerization process can be studied by first developing a theory of the breaking of single molecules and then taking into account the distribution of breaks in the various chains. In such a manner it has been shown3 that if a, the degree of depolymerization, is defined as the probability of a given bond being split (i. e., a is the ratio of the total number of bonds cut to the total number of bonds in the system), the average fraction of monomeric elements existing as components of t-mers in a system that was originally composed of N ( p 1)-mers is
1217
In an almost normal distribution
I
I
I
I
I
I
I
+
F P + l ( P , a) = (1
- CYP
Since the total number of t-mers, N,(fi, a),is given by NdP, a) = NFdP, a)(1 P ) / t ( 13a) implies Nt(p, C Y ) = Na(1 - C Y ) ~ - I [ Z ( p - t ) a ] ,t 5 p (13b)
+
N,+l(P,
01)
=
N(1
- aP
+
350
300
400
P. Fig. 1.-Three
types of molecular size distributions
Since, as one can show by integration by parts'
Now consider the depolymerization of a system that initially consisted of N polymers distributed into Nnl monomers, Nnp dimers, . . , Nn, p-mers, . . , in such a manner that conditions (2) and ( 3 ) remain valid throughout the degradation process. The total number of t-mers, Nt(cr), when the degree of depolymerization is a, is the totality of t-mers generated from each of the original molecular species composed of t or more monomers. Thus m
= Nn, ( I
- 0l)l-l +
iv
np+1a(l
-
01)1-1[2
+ ( p - t)al
P+l=l+1
Let us assume that the initial material has an arbitrary size distribution in which the fraction of p mers is
Then if we let 1 -
CY
=
p and
- .f i.)./ a
= z
Tables of these function.: exist in Jahnke-Ernde, "Fonktiontsfelll," Leipzig, 1033.p. !)S
ELLIOTT MONTROLL
1218
For a given a,the fraction of monomeric elements existing as components of t-mers can be found by substituting (17) into
Average Molecular Weights of the Degraded Material.-The formulas for the distribution of molecular sizes of degraded material that we have derived so far are functions of the initial distribution and the degree of depolymerization, a. In most problems i t would seem that this form of a result is of little value because one usually cannot measure a directly. However, when i t is possible to determine any one of the average molecular weights of the partially depolymerized system, a can be found as a function of that average molecular weight, and then the history of the reaction can be followed by making molecular weight observations a t various stages of the degradation process. The relationship between “number average’’ molecular weight, M:, and a is immediately apparent from the definition of M,*, for M,*/m = average number of monomeric elements in a chain, that is
M,*=
Total number of monomeric elements in the system Total number of molecules in the system
Since the total number of monomeric elements is conserved during the reaction (i. e., bonds internal in monomeric elements remain intact and only bonds connecting monomeric elements are split) the total number a t any time is the same as that in the initial material; that is, x p n , = pl. Now the number of bonds in a ( p 1)-mer is p , thus the number of molecules whose source was ( p 1)mers is ( 1 ~ u p ) n ~ and + ~ ,the total number of molecules in a system with the degree of depolymerization a i s C(I = c[(1 - a) a ( p l)]n*+,= 1 - a apl. Therefore
+
+
+
+
M Y m = M/D
( p 1)-mers, when a and p are sufficiently large so that3 (1 - a), < < a ( p 1) M,/m 1 2(1 - a ) [ a ( l p ) - ll/a2(1 p ) = 1 4- 2(1 - a)/a - 2(1 - a)/a2(1 + p )
-
--
+
+
+
1
+ 2/a:
Suppose p is about 200. 2/a
-
1= 9
+
>> 2(1 -
If a is as little as 0.2 p ) = 1.6/8 = 0.2
a)/a2(1
+
Thus with initial chain lengths > 200, and a > 0.2, M z / m is independent of the initial chain length. This means, when a system with molecules of various degrees of polymerization > 200 is depolymerized until a > 0.2, M$/m becomes independent of the initial distribution and MZ/m -1 f.2 / a (a: > 0.2) (20)
-
the next approximation being obtainable by subtracting 2(1 - a)/a2h1. At low degrees of depolymerization the corresponding derivation is more complicated because one must evaluate directly
using the expression for N f given in equation (17). The value of t2Nt/Nis calculated in the appendix (iii) and its substitution in (21) yields Mz/m =
(22)
When nt = 1 if t =p 1 and . E -_
+
=
p
+ 1 and zero otherwise,
11.1
m
+
+
+
+
VOl. 63
+ a ( m - 1)1
which agrees with the formula for the weight average molecular weight of an initially uniform system. In an almost normal initial distribution
(19)
I n an initially homogeneous distribution with Pl=P+l M$/m = ( p
+ 1)/(1 +
ap)
which checks with the corresponding formula derived in reference 3. To calculate the “weight average” molecular weight, M z , a t high degrees of depolymerization, we make the following observations. In the case of a system in which all initial molecules are
Application of equation (ix) of the appendix simplifies this to m
a
MOLECULAR SIZEDISTRIBUTION IN POLYDISPERSE SYSTEMS
May, 1941
2
US)
exp.
(-pia + T) I (25)
1219
(14), (19), etc. Some other aspects of the kinetics of depolymerization, such as the probability of a given number of bonds being broken in a given time, follow immediately from the solutions of the corresponding problems in the theory of radioactive disintegrations.6 I
I
I
I
I
I
Using equation (19) or (23) one can determine a from a measured average molecular weight (as
discussed in reference 3, p. 724, when M z is the measurable variable, a is most easily found by graphical means) and substitution of the a value into (17) or (18) indicates the molecular size distribution. Figure 2 shows the degradation curves of an initially normal molecular size distribution and emphasizes the transition from one sharp distribution to another. A method to decide whether a given reaction proceeds with random splitting of bonds as discussed in this paper would be to measure both M,*and M: a t various stages of degradation, and then compare the measured M: values with those calculated from (23) using the a values derived from (19). Time Dependence of a.-To express the time dependence of a we shall make the assumption that throughout the depolymerization process the rate at which bonds are broken is proportional to the number of uncut bonds6 I n this case dB/dt = - XB, if B is the number of uncut bonds a t time t and X is a degradation constant. Thus B(t) = Ne-it
(pl
- 1)
- B ( t ) = N ( p l - 1)(1
200 300 400 t. Fig. 2.-Fraction of monomeric elements existing as components of t-mers for various a's in a system that originally had a normal distribution of molecular sizes ( p ~ = 350 and Ms/m = 353).
cz
Calculation of lecular Weights.-From
- e--hf)
N , / N and Weight Average Moequation (14),if (1 - a ) = b
t*
t2Nt/N -t=1
t2ntPt-1
m
but for any functionf(p, t ) (for which the following sums converge) m
c c f(PA m
=
m
1=1 p - 1 a
tzn,~"2 a
The degradation constant, X, for a given reaction can be evaluated by observing average molecular weights at various times, finding the corresponding a's from (19) or (22), and applying
m
m
p-1 1=p
f(PA
- ( t -t 1)a + pa1 =
m
t*B%,(2 121
-
p-t+1 m
m
- ta + pa) -
m
t*n*8' [2 9-1 1-9
-
(t
+ 1) + Pal
m
9-1
(6) See, for ,.,.,.-.
(IYJOJ
pn, = PI. Using the fact that
np = 1;
Now
One can express the size distribution or average molecular weights of a partially degraded system as a function of the time by substituting (27) in ( 5 ) A similar discussion of this problem has been made by W. Kuhn. Bcr., 63, 1503 ( 1 9 3 0 ) .
1
Thus l=l p-r+1
- ( I / t ) log (1 - a)
m
m
Interchanging the order of summation in the second term:
(ii)
A =
+
i=1 a
m
Therefore
Appendix
m
(i)
(26)
since the total number of bonds in the undegraded system (t = 0) is N Z r t p ( P- 1) = N(p1 1). Now the total number of bonds split a t the end of time t is B(0)
100
0
P"1
example. Ruark and Devol. Phrs. Rev., 49, p. 355
1220
ELLIOTTMONTROLL
we have
Letting c =
PI
+ u2 log /3 and x = p
m
m
1=p
t3p' =
-c
m
--e-a)' + p(1 + ap + 3pz -
(1 -
3pp)
($9
+p
+
y ~ ~ p * 3pa)
+ p3(1 - p)zi
Substituting these expressions in (ii), and (ii) in (i)
PNt/N =
(iii) 1=1
M(2
- a ) / a - 2 (1 -
2
npPP)(l
- a)/d
p=1
For degrees of depolymerization sufficiently small to make ( a p l ) amuch less than apl nPpp =
=
1-a
+
p ~
n, (1 a2(p2
- ap
Thus the weight average molecular weight of an initially almost normal system is (ix)
+ alp(# - 1)/2 - . . . )
AI: Til UJ
- $1)/2! - a3(p3 - 3p2 + 2 / ~ 1 ) / 3+! . . .
( 2 - a ) - 2(1 - a) -11 t a
e2
+0, log (1 - a ) u: 2 - a (1) ---
a s a
and m
ffl
----
If a is small enough to make the use of this sum necessary, we can replace the summation operation by a n integration, and since exp. 1. - (PI - p ) z / 2 u 2 ) + 0 very rapidly as p becomes small, we can take the limits of intem, Therefore gration as - rn and
+
+ i: log3 (1
-
[
-a, and
- 2.3)
esp.
(-PI0
+ a W / 2 )!I
Summary
So (iii) and (iv) imply
Replacement of the p's by the corresponding functions of average molecular weights describes M ~ / mas a function of the degree of depolymerization and of the initial average molecular weights. In a n initially almost normal distribution
-
[I
a
(1
and for very small a
I
Pla2
+ p l l u y (1 -a)
Iog2(1--(1~
In a polydisperse system of long chain molecules of the same general structure with a single maximum molecular size distribution, this distribut'on will not deviate radically from a normal one. With this in mind a general distribution function (which shows deviations from normal) is derived in terms of three measurable average molecular weights. A theory of depolymerization of an arbitrary distribution of high polymers is developed statistically under the assumption that all bonds con necting monomeric elements in the system have the same probability of being broken. The molecular size distribution a t any time is given as a function of the initial distribution and of the fraction of bonds split. Under the assumption that the rate a t which bonds are cut is proportional to the number of uncut bonds in the system, the time dependence of the degree of depolymerization is discussed. NEWHAVEN,CONN.
RECEIVED JANUARY 4, 1911
