THOMAS H. DONNELLY
1862
The Direct Estimation of Continuous Molecular Weight Distributions by Equilibrium Ultracentrifugation
by Thomas H. Donnelly Swift & Company Research Laboratories, Chicago, Illinois 60609 (Received December 3, 1966)
The principle of direct determination of molecular weight distributions by equilibrium ultracentrifugation was established long ago. The present work formalizes this principle by expressing experimental results in Laplace-transformable functions, and by determining molecular weight distribution functions as the Laplace transforms of these functions. This technique gives further insight into the types of molecular weight distributions which cannot be distinguished from each other by equilibrium ultracentrifugation.
Introduction The possibility of using measurements of the redistribution of solute species a t ultracentrifugal equilibrium for the complete description of molecular weight distributions of polydisperse solute systems has been discussed many times since the technique was first developed.2 The practicability of using this technique and some of the limitations of its use have also been inve~tigated.~The present study represents another attempt to describe both the theory and practice of the use of equilibrium ultracentrifugation for characterizing molecular weight distributions on a more quantitative basis by expressing experimentally observable data in terms of closed functions from which apparent distributions may be readily calculated. Theory This study is a continuation of work previously reported4 for discrete, limited distributions of virtually uncharged solutes. Previous studies5~6 have suggested that solution nonideality and variation of density with concentration should be handled by investigation at or near the Flory temperature and extrapolation’ to zero concentration, while compressibility of solute and solvent species may be formulated in terms of adiabatic compressibilities of the species. As an equation to be satisfied a t zero concentration, we may write d In N t
=
M,(1 - @,p)Adu
where The Journal of Physical C h m i s t r y
(1)
A =
- x m 2 ) - w2b 2RT RT
w2(xb2
and (3)
In reasonably dilute solutions, such as those generally used in ultracentrifugation P = PO
+ aw
(4)
(1) Presented in part before the Division of Agricultural and Food Chemistry a t the 148th National Meeting of the American Chemical Society, Chicago, Ill., Aug 31, 1964. (2) See, for example, H. Rinde, Dissertation, Upsala, 1928; E. 0. Kraemer in “The Ultracentrifuge,” T . Svedberg and K. Pedersen, Ed., Clarendon Press, Oxford, 1940, p 342 ff, reprinted by Johnson Reprint Corp., New York, N. Y., 1959; R. J. Goldberg, J . Phys. Colloid Chem., 57, 194 (1953). (3) See, for example, M. Wales, F. T. Adler, and K. E. Van Holde, ibid., 55, 145 (1951); J. W. Williams, K. E. Van Holde, R. L. Baldwin, and H. Fujita, Chem. Rev., 58, 715 (1958). (4) T. H. Donnelly, J. Phys. Chem., 64, 1830 (1960). ( 5 ) L. Mandelkern, L. C. Williams, and 3. G. Weissburg, ibid., 61, 271 (1957). (6) H. Fujita, A. hl. Linklater, and J. W. Williams, J. Am. Chem. Soc., 82, 379 (1960). (7) One aspect of the procedure for extrapolation proposed in our previous study4 can be shown to give rise to anomalous results. This is the use of (Zb2 - z,2)ref as proposed. For a single solute species, it can be shown that F(n,z),as defined in eq 18 of the present work, is invariant in (Zb2 - z,2), and can thus be extrapolated directly at constant temperature and revolutions per minute. For a multicomponent system, no such procedure of extrapolation is valid unless F(n,z) is essentially invariant in A as defined in eq 2 within the range of A encountered. The value of A to be used in estimating molecular weight distributions would then presumably be the average value for the several determinations.
NOLECULAR WEIGHTDISTRIBUTION BY EQUILIBRIUM ULTRACENTRIFUGATION
where w is the weight fraction solute, and a is a proportionality constant. It can be shown that, if iii is independent of i for all solute species (1 - ap) =
a AM u
s,..-. _-
M $ ( M )e
(518
P
(14)
- 1)
(e
On extrapolation, this becomes
7dM
(e w - 1) M$(M)dM
noun2
a(1 - w)
1-gp=-
1863
This may be integrated to give P m
Cl
s,'
PO
Thus, eq 1 becomes
du
PO = -
aA
M$(M)dM (e
aA d In N i = -M,du
(7)
PO
- 1)
PO
Since # ( M ) is such that
This gives
lm$(M)dM= 1
(16)
eq 14 becomes Within the condition that the solution is infinitely dilute and the solvent and solute are equally compressible, the condition of conservation of mass becomes L I N u , du = constant
(9)
Thus
a A Mu
g=@J1?$dul nco PO 0 nc
M $ (aAM M ) e T dM
(17)
- 1)
(e
The purpose of the present study is to determine the manner in which the experimentally measureable function, n:/n2, should be expressed in order to obtain ready evaluation of the molecular weight distribution function, $ ( M ) . In this work, one appropriate procedure is the use of the function _.
b
.
Therefore x \dx)
a z u
n
i=O
aA
NuiMi = -
C
PO i = l
e
(Ni)oMi2
m
(11) (e
\
If this is done, eq 17 may be written as
- 1)
For constant refractive increments, refractive index differences may be written as
By a change of variable such that
aAM --
i=O
- t
PO
Then, under the conditions of eq 9 eq 19 becomes u & a
(21) For continuous distributions, this becomes
(8) This is equivalent to the similar equation of M. Wales and S. J. Rehfeld, J . Polymer Sci., 62, 179 (1962).
Volume 70, Sumber 6 June 1966
THOMAS H. DONNELLY
1864
If we take
P=
RT
dM(1
- ep)
If and 1
u
then
K = ( S ) L %du eq 21 becomes
This can be evaluated as 1 By taking l - u = s eq 24 becomes
(Q
Equation 26 evaluates an integral on the left which is of the form of a Laplace transform. Thus, if F(n,z) can be evaluated from the experimental data so that the mathematical operations on the right of eq 26 yield functions which can be expressed as Laplace transforms, then +(t) = $ ( M ) can be evaluated. The simplest case to be evaluated is that in which F(n,z)
P
=
(27)
This is obtained from ideal, single solute systems. Here
b
($ - 1 +
S)
(34)
In this case, it can be shown, as is done in Appendix I, that
Higher polynomial expressions for F(n,z) do not lead to readily evaluable Laplace t'ransforms, so that it is not appropriate to fit curvilinear plots of F(n,x) with these functions. Such behavior can often be handled by F(n,z) =
bu
)' b
P
1 -
bs
P - QU 1 R(P - Qu)
+
Here =
The function whose Laplace transform is is the unit impulse function, or the Dirac 6 function of ( t - b / p ) , 6(t - b / P ) . This gives h
- exp K
This can be evaluated to give
1
Since
lm
6 (t - i ) d t = 1
The Journal of Physical Chemistry
b(1
+ R ( P - Qu))du P - QU (37)
i" tdt)e-"dt
this can be shown to be a distribution concentrated at t = b/P. This is equivalent to previous formulations, since it gives
S,
Here it can be shown that
ebRu
1865
R~OLECULAR WEIGHTDISTRIBUTION BY EQUILIBRIUM ULlTRACENTRIFUGATION
4(t) = 0; 0 < t
< bR
(39)
t 2 bR Experimental Section This procedure has been tested experimentally by application to many varied samples, in none of which the molecular weight distribution was known from other measurements. As an example, samples have been prepared from Swift's "Superclear" gelatin, an acid-cured porkskin gelatin, by degradation at 95" a t pH 4.0. Schlieren patterns for two of these samples, run at a concentration of 0.5% at 40" in pH 8.6 0.1 M sodium barbiturate buffer, are shown as A and B of Figure 1. A represents a sample degraded at 60" for 40 hr, while B represents a sample degraded for 31 hr at 40", these particular samples being chosen to illustrate different types of behavior. Figure 2 shows F(n,z) for these samples and gives some idea of errors of measurement involved in evaluating this function. The values of (nJo4are evaluated by interference optics and converted to the same units as those in which dn,/dz is measured. This conversion also involves the units of measurement of radii of rotation, foi. which any convenient units may be chosen. The magnitude of P , &, R , b, and F(n,z) will, however, also depend on this choice of units. While centimeters may be rational units for presentation of data, other units, such as those of the measuring micrometer, may be more convenient for the collection of data. For the
samples presented here, dnJdx is measured using a Nikon Shadowgraph Model 6 with a metric stage having a vernier unit of 0.002 mm. One millimeter on this micrometer is taken as the unit of radius at standard magnification. The calculations associated with studies of this sort are best carried out using an elcctronic digital computer.* Table I shows the input data sheet for the sample of Figure 1A. This is basically a table of dnJ dz, or F(z), us. z, with the number of observations Table I: Data Sheet for Sample of Figure lA, Showing Data Gathered far Electronic Data Processing* F(Z)
1:
140.700 140.892 141 ,084 141 , 2 7 6 141 .468 141 , 6 6 0 141.852 142.044 142.236 142,428 142.620 142.812 1 4 3 .o n 4 143.196 143.388 143.580 143.772 143.964 144.156
2.177 2.341 2.442 2 . 5 7 1
2 2 3 3 3 3 4 4 4 4
,735 .905 .089 .330 .550 .801 .022 .372 ,5 5 6 .873
5 . 2 5 9 5.7351 6 . 2 7 5 7 , 186 7 ,!IO6
Wnit are those of the measuring micrometer. Number = = 0.295, and (n.)o = 19.0, A = 0.~00566,w = 0 . ~ 5 0 ,
23.83.
given. These are generally collected by measuring the interval between the meniscus and the base of the spinning column and dividing this into suitable intervals. The z-micrometer dial is then set at the proper value of z and dn./dz is determined using the y micrometer. w is the weight fraction of solute. ( 7 ~ is ~ )the ~ value of (n& referred to above. A and alpohave been defined in eq 2 and4. Figure 3 is a reproduction of results produced during the calculation for this same sample, including the input data. Because the printer uses only capital
Figure 1. Schlieren patterns irom two repmentative samples at equilibrium at 10,889 rpm; a 0.1-ml sample WBS used with 0.05 ml of 3 M Fluomchemical FC43.
(9)The calculations presented here were carried out using an IBM 1620-40K. A program suitable for this has been written in P O R T R ~ N 11. Version 2. and is available on request. Desk calculators such as the Underwod-Olivetti Tetraotys or Divisumma GT are suited for hand ealculntions. but these are likely to require several hours rather than B few minutes of electronic data processing time needed.
Volume 70, Number 6 June I966
THOMAS H. DONNELLY
1866
Figure 2. Plots of F(n,z) us, u for the samples of Figure 1.
Roman letters, many quantities are shown as an approximation of the symbols used in the present work. Thus, (n,)~is NCO, w, the weight per cent solute, is W, alp0 is A/P. A is A, defined by eq 2, and B is the b from eq 2. NUMBER is the number of s,F(z) pairs read. P, Q, and R are the constants of eq 38 as evaluated. GAMMA is I?@/&). Z, XX, and V1 are constants derived from the data by the procedure of Appendix 11. V is equal to V1, unless V1 is not between 0 and l, in which case it is taken as 0 and a straight line is used to fit the data and estimate the distribution. X is z, FX is (dn,/ds), and U is u of eq 3. FOX is l/z(dn,/dz). NCU is (n,") computed as in our previous work.4 FNX is F(n,z) from eq 18. B/FNX is (d In n,/du) by eq 18. PSIM is $(t), as in eq 19, 22, 24, etc. CHIM is l : ( t ) d M .
I,
- bR, and M
all contain an approximate correction for the variation of (1 - aP> with concentration. This is based on eq 5 using wu/wO = ~ c ' / % O and leads to the t
definitions u' =
- (1
du' -= 1 du
E)
-I-
wo
" n% n,O
- (1 + ); wonc" n,O
The Journal of Physical Chemistry
(40) (41)
1' is u' where u = 1. A series of approximations of the order of w2 = 0 gives t =
(1')aAM PO
(42)
and
for eq 20 and 23. Because of this, I is the integral in eq 43, and M is (pot/(l')uA). T - BR is t - bR and is used as an indexing variable. Figure 4 shows CHIM calculated by this procedure for the samples presented in Figures 1 and 2.
Discussion The technique presented here seems to provide the most mathematically rigid method available for the direct estimation of molecular weight distributions from measurements on systems at ultracentrifugal equilibrium. The principal assumption made is that ths distribution is continuous, which makes the whole procedure possible. It should not be used if better assumptions can be made, but provides a good picture of molecular weight distributions in cases where the transformation from eq 13 to eq 14 is reasonable.
~IOLECULAR WEIGHTDISTRIBUTION BY EQUILIBRIUM ULTRACENTRIFUGATION
1867
OUTPUT MOLECULAR WEIGHT O I S T R I B U T f O N 17 MAR 1965
E-323 J-43 X
140.7Oooo 140.89200 141.08400 141,27600 141.46800 141.66000 1 41.85200 142.04000 142.23600 142.42800 142,62000 1 42.8 1 200 143.00400 1k3.l%QO 143.38800 143.58000 143.??ZOO 143.%400 144.15600 NUHBER
t9.
.
z
552.96056 P
,11574E44 T-BR
.025
.050
.loo .zoo
.150 .250 .300 350 ,400 .450 .500 550 .6m .650 .700
.750 .800 ,850
.9eo
.950 1 .Ooo 1.050 1.100 1.150 1.200
1.250 1.300 1.350 1 .400 1.450 1.m
1.550
1.600 1.650
1 .700
1.750 1.800 1.850 1.900 1.950 2.000 2.050 2. Hwt 2.150 2.200
2.250 2.300
2.350 2.450
*.-2
2.508
FX
2.17700 2.34H)o 2.44200 2.57 100 2.73500 2.90500 3.08900 3.33000 3.55000 3.80 100 4.02200 4.37200 4.55600 4.87300 5.25900 5.73900 6.27500 7.18600 7.90600 A
.00005660000
U
.O 1547 .O 1661 .O 1730
* 55255
.02820
.60822 c 66397 .71978 -77568 ,.E3164 .E8769 ,94380 1 .ooooo NCO
23.83000
xx
1157.44433 Q .55296E+03
V 0~0000000 B
.01819 -01933 .02050 .02177 .02344 ,02495 ,02668 ,03061 .03185 ,03403 .03667 .o 3997 .04364 .04991 .OS484 W
FNX
1173.69304 1119.06186 1100.76748 107 3.40335 1036.75667 1003.81 299 971.72303 928.90 2 10 898.97666 867.18855 847.27346 806.8 1 602 802.17457 777.59698 748.00851 71 2.77796 679.18989 619.76595 590,49192
A/ P
29500
v1 -.0417318
0.0000000
R I
.10766E+01
.13042€+01 CHlM
PSlM
.0000326408 ~ 0 0 0 29694 0 .0000248837 .0000219891 .MX)O1%882 .OOOO 177572 . ~ m 9 0 9 .0000146291 .0000133331 .OW0 1 21 754 .0000111358 .0000101982 ~OoooO93500 .0000085807 OoooO788 14 ;0000072447 .0000066640 .00000613 7 .OOOOO564~9 .0000052052 .000004 988 .OOOOO d 2 6 2 .0000040844 .OW0037706 -00000 34823 .OOOOO32 172 .OooOo29735 .00000 27491 .0000025426 .OW00 23524 .000002 1770 .0000020154 .00000 18663 .0000017288 .00000 1601 9 .0000014846 *.oooM)13764 ..0000012763 .00000 1 1839 .0000010984 .0000010193 .0000009462 A"8785 ;0000008158 .0000007578 .0000007041 .0000006543 .0000006082 .0000005654 .OW0005258 .0000004890
.
NCU
18.16012 18.59384 19.0 5300 19.53424 20.04361 20.58503 21.16045 21 .??666 22.437 12 23.14280 23.89380 24.69960 25.55667 26.461 84 27.43449 28.49026 29.64357 30 93577 32.38456
.00500
GAMMA
.49223E+03
M
1507.03920 3014.07841 6028.15683 9042.23525 1 2056.3 1367 1 5070.39209 18084.4MSt 21098.54893 2411 2.62735 2?126.705?? 3i)w;i 8 4 i 9 33154.86261 36 M8.94103 39183.0 1945 42l97.09787 45211.17629 48225.25471 51 239.33313 54253.41155 57267.48997 6028 1.56839 63295.64681 66309.72523 69323.80365 72337.88207 75351.96048 78366.03890 81380.11732 84394.19 74 87408.27h6 90422.35258 93436.43100 96450.50942 99464.58784 102478.66626 1 05492.7 4468 108506.823lO 1 1 1520.90152 114534.97994 117549.05836 120563.13678 1 23577.2 1520 126591.29362 129605.37204 132619.45046 13 6 3.52888 1 5617 60730 l446p:l62?+ 1$1661:68 72 1476 9 4256 1 50 P3 92097
FOX
0 .OOOOQ ,05491 ,10991 .16498 .22012 -27534 33063 .38600 .44144 .49696
,0762449079 ,1227440301 .2040544526 ;2?4693?i94 .3375031 340 .3939349532 .4449454465 4912418372 ;5333821549 .5710246482 .GO69556653 .6391069298 .6685670307 .6955894983 .7 203987344 .7431945251 7641555742 .7834423375 .8011993402 .8175571017 .e326337580 .E465364429 -8593624749 .87 12003876 .8821308266 .8922273309 9015570672 ,9101813665 .9181563512 .9255333840 .9323595124 .9386778613 .9445279847 .9499461836 .954965?916 .9596174344 .9639292635 .9679271680 .9716349708 .9750745960 .9782662357 .9812284909 .9839785041 .9865320798 .9889037941 .9911070949 .9931543934 .99505?14?9 .9682594O3 .9904705465 1.0000000000
-
B/FNX
.41938662, .43986054 ,44717088 -45857055 ,47477983 .49036142 .50655500 .52990639 ,.54754609 .56761723 .58095902 .61009096 .6 1 362100 .63301578 .65805556 .69058134 -72473276 .79422104 .E3359509
.
Figure 3. Processed data for the sample of Figure lA, as printed by printer from punched cards produced from calculation using program mentioned in footnote 9. Symbols are explained in text.
Volume 70,Number 6
June 1966
THOMAS H. DONNELLY
1868
distribution of two components in an ideal solution. This may be represented as the combination of two distributions such as represented by eq 28 and 29. Table I1 gives the results of a calculation of F(n,z) for such a Table 11: Comparison of F(n,z) for Tv.-o-Component and Continuous Distribution
--
loo:ooo
50,000
0
MOLECULAR
150,000
WEIGHT
1.0
: 2
08
c
E0
a
06
F(n,z)
F(nA
11
F(n,z)
(eq 36)
(eq 32)
0.0 0.1 0.2 0.3 0 4 0 5 0 6 0 7 0 8 0 9 1 0
1.5000 1.4875 1 ,4750 1.4625 1 4502 1 4378 1 4255 1 4134 1 4013 1 3894 1 3775
1.5000 1.4874 1.4749 1.4625 1 4501 1 4378 1 4256 1 4135 1 4014 1 3894 1 3774
1.5000 1 ,4870 1.4748 1 ,4626 1 4504 1 4381 1 4259 1.4137 1 4015 1 3893 1 3771
-I
c 0 I-
0 4
0
IL
z
p
02
0 4 E
I 00 0
I00,000
50,000 MOLECULAR
150,000
WEIGHT
Figure 4. Integrated molecular weight distribution functions for samples of Figures 1 and 2 as determined by-procedure of text.
case. The distribution is taken to consist of two parts by weight of a given molecular weight material and one part of a material twice as large, run at some appropriate speed. b is taken as 1 em2. Values of P are taken as 2 and 1 cm2. These might correspond to molecular weights of 4000 and 8000, respectively, at 20,000 rpm. Column 2 of Table I1 gives the theoretical values of F(n,x) from the equation P.
L l
F(n,x) = The assumption that specific refractive increments are constant is one that recurs throughout molecular weight studies. While this assumption is made to derive the formulations given here, a systematic variation of specific refractive increment with molecular weight could be used to derive related formulations. For example, this parameter could be incorporated into t of eq 20. Variation of 8, with i should be capable of similar treatment. It is possible to take into account some small variations of density with radius of rotation by using the integrated expression for noU/noO to give w a t any point in the cell, as done in eq 40-43. This leads to integrals which can only be approximated, especially when eq 36 is appropriate. Presumably, an analogous approach could be used to allow for small changes in activities. However, the use of ideal or nearly ideal solutions is probably a more rational approach. The major limitation of this technique is one which becomes apparent by a consideration of a hypothetical The Journal of Physical Chemistry
1 +
(P1- Pz)
(44)
P1-Pa
which can be derived from eq 28 using c parts of PI and d parts of P2. Column 3 is derived from this data using the procedure of Appendix 11. Column 4 is a least-squares straight line, evaluated by a method equivalent to that of Appendix 11 where R = 0. If it is assumed that F(n,s) in column 2 represents the true values, it is apparent that both columns 3 and 4 agree with these values to well within usual experimental error, especially that of measurement of (dn,/ ds), Since column 2 represents a model of the sharpest possible bimodal distribution, and since other bimodal distributions will only blur this picture, this calculation has grave implications with regard to the resolving power of this technique in the study of molecular weight distributions. This situation can be shown to persist with broader differences in molecular weights of the species, although the values of the corresponding dif-
MOLECULAR WEIGHTDISTRIBUTION BY EQUILIBRIUM ULTRACENTRIFUGATION
ferences between column 2 and column 4 show increasing magnitude, the value for column 4 being as great as about 2% less than that for column 2 at the end points, and 1% high in the center, for a distribution of equal parts of two species, one ten times the molecular weight of the other, at the proper operating conditions. It is thus possible that bimodal distributions may be detected by systematic deviations of the experimental data from the function calculated to fit it. Until reliable procedures for carrying out calculations designed to detect such bimodality can be devised and errors reduced, it can be said that this technique will produce only unimodal representations of molecular weight distributions. In addition to demonstrating its small deviation from column 2 for this model distribution, column 4 is given since the value of R used to generate column 3 is negative. This gives a meaningless picture of the distribution because of the conditions used to derive eq 39. I n cases where R is negative, such a procedure must be used in order to obtain a physically meaningful representation of the distribution. n%/n2 can be measured much more accurately than (dnC/dz).l0 It might be proposed that this be used directly to estimate the distribution since eq 19 is really the pertinent equation and F(n,z) serves only t o aid in its evaluation. Indeed, the error in measurement of ncu/n$is of the order of the deviation of the figures in column 4 of Table I1 from those in column 2.1° However, examination of the values of NCU and FNX in Figure 3 shows that the latter is a more sensitive function of u. This is further borne out by reference to Table 111. This table is constructed from columns 2 and 4 of Table I1 by evaluating the hypothetical values of neu/n$which would be consonant with the given values of F(n,z). Inspection of Table I11 shows that the order of magnitude of the differences in n,"/n$ is sufficiently less than that of the differences ~~
~~
~
Table I11 : Evaluation of nou/ncOfor Hypothetical Distributions U
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8 0.9 1.o
From col. 2
From col. 4
1.ooooo 1.06924 1.14391 1,22450 1.31154 1,40559 1.50727 1.61729 1.73639 1.86541 2.00524
1.ooooo 1.06929 1.14397 1.22457 1.31161 1.40566 1.50732 1.61731 1.73640 1.86540 2,00524
1869
in F(n,x) that the improvement in evaluation of this integrated function does not improve the estimate of the molecular weight distribution. In addition, it should be pointed out that methods for evaluation of P , Q and R from values of ncU/n2are not readily available if it is proposed to use functions such as given here for estimation of distributions by Laplace transformation. It might be proposed that n,"/n$ be expressed in terms of other Laplace-transformable functions. Since u goes from 0 to 1, it might be possible to reflect the function around the y axis and evaluate any set of data as a Fourier cosine series. However, neither sines nor cosines are Laplace transforms for any known function. The same is true for a power series in u,although a series in reciprocal powers of a binomial containing u might be an appropriate manner of expressing n%/ncO. This procedure, however, is likely to founder on the same objection as that encountered by Wales, et u L , ~ and others, in that the ". . . .distribution may still be negative in some regions." This type of behavior will not be observed for the distributions estimated by the present procedure. Other procedures for the estimation of molecular weight distributions include gel permeation chromatography" and measurement of sedimentation coefficient distribution.12 Both of these techniques seem to be superior in resolving power to equilibrium ultracentrifugation, unless the observations made here are revised by future developments. However, both also have relatively severe limitations. I n the case of gel permeation chromatography, the most severe limitation stems from incomplete rigorous theoretical explanation of the process. This necessitates calibration with materials of known molecular eight.'^ A similar end might be served by determining molecular weights of eluent fractions by a procedure such as equilibrium ultracentrifugation. A recent summary of the status of determination of molecular weight distributions by sedimentation velocity has been given by Blair.I4 It is to be supposed that bimodal distributions giving two sedimenting peaks could be resolved by this technique. However, once again this is a technique in which sedimentation constants must be related to molecular weights, and is further complicated by distributions of diffusion constants. (10) E. G. Richards and H. K. Schachman, J . Phys. Chem., 6 3 , 1578 (1959). (11) J. C. Moore, J . Polymer Sei., A2, 835 (1964). (12) UT. B. Bridgman, J . A m . Chem. SOC.,64, 2349 (1942). (13) L. E. Maley, J . Polymer Sci., C 8 , 253 (1965). (14) J. E. Blair, ibid., C 8 , 287 (1965).
Volume 70.Number 6 June 1966
1870
THOMAS H. DONNELLY
It may thus be said that, of these three methods for estimating molecular weight distributions, the technique based on equilibrium ultracentrifugation has the most rigorous theoretical basis, but that it is severely handicapped by likely experimental inaccuracies and the relative insensitivity of a single estimation to multimodality of distributions. Thus it can be said of the estimations of distributions produced by this technique that such distributions would give experimental behavior in ideal dilute solutions agreeing with the observed behavior of the sample within the inaccuracies cited. Although the methods of gel permeation chromatography and of sedimentation constant distribution are more dependent on molecular shape than the equilibrium method, their straightforward advantages in resolving power suggest that they may be profitably used in conjunction with the equilibrium method. The most definitive procedure for study of molecular weight distributions today might thus involve fractionation and empirical study by gel permeation chromatography coupled with the absolute determination of molecular weight and homogeneity of fractions by equilibrium ultracentrifugation in ideal dilute solutions. Acknowledgments. This work could not have been done without the skillful technical assistance of A h . Peter Arendovich. I n addition, Mr. David Tierney contributed extensively to the writing of the computer programs used. The mathematical aspects of the problem have been discussed profitably with Dr. Kenneth Hanson. We are also grateful for having had the opportunitg to discuss this work with Professors J. W. Williams and I
