A general method for the determination of molecular weight distributions by dynamic light scattering

Eur. Polym. J. Vol. 18, pp. 847 to 861. 1982 Printed in Great Britain. All rights reserved

0014-3057/82/100847-15503.00/0 Copyright © 1982 Pergamon Press Ltd

A GENERAL METHOD FOR THE DETERMINATION OF MOLECULAR WEIGHT DISTRIBUTIONS BY DYNAMIC LIGHT SCATTERING J. RACZEK Institut for physikalische Chemie der Universit~it Mainz, Jakob-Welder-Weg 15, D-6500 Mainz and Sonderforschungsbereich 41, Chemie und Physik der Makromolekiile, Sektion Mainz, Germany (Received 25 January 1982) Abstract A procedure is described for the determination of the molecular weight distribution (MWD) by dynamic light scattering, in which no restrictions with respect to the scattering system are necessary. The procedure involves the construction of one single curve from a .system of experimental curves. This curve is suitable for the determination of the MWD since all effects are eliminated which are in general not exactly describable by theory (such as concentration effects, intramolecular motion, scattering function) and which therefore can systematically distort the results for the MWD. The procedure is illustrated and tested for a polystyrene with Mw -~ 18 106, in toluene at 20~C.

INTRODUCTION

In recent years, interest in the use of light scattering for the determination of the molecular weight distribution (MWD) has increased due to inter a l i a - several advantages associated with this method. The problem of the determination of the M W D from experimental curves obtained from classical light scattering (CLS) was first formulated theoretically by Greschner [1, 2] and discussed in detail for Gaussian coils. Meyerhoff et al. [3], Lechner [4] and Hack and Meyerhoff [5] have carried out determinations of MWD-parameters for this particle shape using the scattering-function of the CLS. Several analytical procedures for the experimental curves obtained by dynamic light scattering (DLS) have also been described, which can be used for the characterisation of the M W D of scattering particles [6-18]. However, these procedures are in general only of limited application if intramolecular interferences must be taken into account. Thus the molecular weights of the individual species must not be too large. Furthermore, when the extension (i.e. the radius of gyration) of the scattering particles is insufficiently small compared with the wavelength of the light used, then the scattering-function of the corresponding particle shape must be known. The procedures are also limited to scattering systems in which the concentration effects on the scattered light are small; however, even under this condition possible sources of errors in the determination of the M W D are still involved, when these small concentration effects are neglected or treated purely empirically (as discussed [19 22]). These difficulties could to date * In the present work, the symbol C for the autocorrelation functions relating to one recording technique will be indexed with "ht" for the heterodyne- and "'hm'" for the homodyne-technique respectively. Relations which are valid for both recording techniques will not be indexed. The frequency spectra (symbol S) are treated analogously. 847

only be overcome if one was content to use averages of the coefficients of diffusion and/or friction etc. for the characterisation of the MWD. The present work develops a procedure for the determination of the M W D from experimental DLScurves in which no restrictive assumptions concerning the scattering system are made; all effects are eliminated which cannot exactly be described theoretically and consequently could introduce systematic errors into the calculation of the MWD. A short treatment of this problem has been given [23]. BASIC EQUATION FOR ACFs

Experimental autocorrelation functions (ACF) of DLS--designated by C e x ~ e p e n d on solution properties and also on the (correlation delay) time t, concentration c and the scattering parameter k = (4 7r n~)/Ao" sin(0/2) (nB: refractive index, 2 o : wavelength of the primary light source in vacuo, 0: angle of observation). The curves are furthermore affected by procedural and instrumental parameters, such as measurement-duration of individual curves, intensity, coherence area, clip-level (in the case of clipped ACFs) and so on; dependences on these latter parameters will be designated in the present work by (.. 3. The experimental curves can always be formally described by the expression* C~x(t, k, c , . . . ) = A(k, c . . . . )" Ci(t, k, c) + B(k,c,...).

(1)

In this relation the term Ci is the signal-term; Ci alone contains information on the scattering particles and their solution properties. The procedural and instrumental parameters (...) influence the experimental curves only via the coefficients A and B. Two principal difficulties are now apparent: (1) The signal-term Ci in Eqn (1), which alone can be used for the determination of the M W D , is in general unknown. (2) In general there are no sufficiently accurate

848

J. RACZEK

expressions for the coefficients A and B; these must accordingly be also assumed to be unknown.* The latter difficulty (2) can be removed--initially on a purely formal basis--by replacing the experimental ACF Cex by a systematised autocorrelation function Cs of the form

Cs(t , k, c) = As(k, c)" Ci(t, k, c) + B~(k, c),

(2)

where the signal-term C~ is identical with that of Eqn (1). The new coefficients A~ and B~ are dependent on k and c, but not on the procedural and instrumental parameters (...); in order to make a determination of the M W D possible, however, these coefficients must be chosen such that they allow a comparison of individual autocorrelation functions C~(t, k, c) at different values of k and c. The following section describes how the transition from Eqn (1) to Eqn (2) may be achieved. Equation (2) for the systematised A C F can now be used to overcome thefirst mentioned difficulty (1), in that an ACF is constructed with a sufficiently simple and suitable signal-term C~. In this context the influences of the correlation of distinct particles and also the effect of the particle shape on the scattered light must first be discussed. The influence of the correlation of distinct scattering particles on the signal-term C~ is directly related to the concentration effects (mentioned in the introduction) and is only with difficulty--/f at all-described by a theoretical treatment. The extent of this influence can be essentially perceived in the term 2A2 Q(O) c of the Zimm-Debye-Equation [-24,25] valid for the CLS (A2: second virial coefficient, Q(0): intermolecular interference scattering-function [-25, 26]. Furthermore, the correlation of distinct particles influences the concentration dependence of the diffusion coefficient. All these (unsuitable) concentration effects are accordingly eliminated by extrapolation of the ACF Cs to infinite dilution (c---* 0). If the scattering parameter is in addition sufficiently small, then the limit of the systematised ACF (2) for any arbitrary particle shape is given by

The signal-term in Eqn (3) is still not sufficiently simple for the determination of the MWD. In general, it is not known a priori how small the scattering parameter k must be, since the values of k which allow Eqn (3) to be applied depend on the polymers in question. Moreover, the scattering-function P(k,M) is exactly known for only a few particle shapes. These difficulties may be overcome by use of the limit k--~ 0, since for all particle shapes the relation lim P(k, M ) = 1

(4)

k~O

holds. It is clear that for this limit (i.e. k---, 0) the time t in Eqn (3) may no longer be regarded as an independent variable; since otherwise for all times t > 0 the constant function lim C,(t, k, c) = A,(0, 0) + B~(0, 0)

(5)

c~O k~O

would be obtained. This means that not only the scattering-function is eliminated, but all other information on the scattering-system disappears also. Accordingly, in order to use Eqn (3) to arrive at an expression which contains the M W D but not the scattering-function, the argument kZD(M)t of the exponential function must be rearranged in such a way that the exponential function in the limit k--~ 0 does not degenerate into the constant 1. This is achieved by the introduction of the variable t': = k 2 t,

(6)

which represents a generalised time of the optical system. t' corresponds to the time t given in units of k- 2 and thus must be redefined for each new value of k'. As consequence, also for k ~ 0 the generalised time assumes all values t' > 0, provided that the time t tends to infinity as t oc k-2. The variable t' expresses the well-known practical experience that for diminishing values of the scattering parameter k the time t must be increased, if comparable segments of ACFs (e.g. in the three-fold half-width) are to be recorded. On substitution of t' into Eqn (3) the limit lim C~(t', k, c)

lim C~(t, k, c)

¢~0

c~O

+ B~(k,O), (3)

+ B~(k, 01 (7)

where n = l for the heterodyne- and n = 2 for the homodyne-technique holds. M is the molecularweight, M,, is the weight-average of the molecular weight and f ( M ) is the molecular weight distribution of the particles. P(k, M) is the (intramolecular interference) scattering-function, which depends on the scattering parameter, the molecular weight and the shape of the scattering particles (not indicated in P). D(M) is the coefficient of the translational diffusion of the centre of mass at infinite dilution.

is obtained. For all scattering parameters k > 0 the integral-transform (7) has the same informational content as Eqn (3). In contrast to Eqn (5), the double limit C(t'): = lim Cs(t', k, c) c~O k~0

= A~(0,0)" / ~ a

Mf(M)e

mM)t'dM + Bs(0, 0)

* Recording techniques and correlators are available which give data-curves with sufficiently accurate values of B (= baseline).

(8)

yields an integral-transform from which the M W D can be calculated. The signal-term in this case is de-

Determination of molecular weight distributions

849

Cs(t.k,c=O)

1,0

0,5

0

0

2,5

5

7.5 I0 nt [IO-Ss]

12.5

I

2

(a)

3

.,, bO'~m%]

I(b) o

Ii

r

I\ i i

a

o

I

2

3

°o

z.

,~I~[~o%']

(c)

5

Io

15

20

~/o 1,0-'~s '1 (d)

Fig. 1. (a) Systematised ACFs C s at vanishing concentration (c = 01 in dependence on t and k. for the example described in the text 01 = 1 for the heterodyne-, n = 2 for the homodyne-technique). (b) As Fig. la, but in dependence on t' and k. (c) Systematised FSs S~ at vanishing concentration (c = 0t in dependence on ¢o and k, for the example described in the text (n = I for the heterodyne-, n = 2 for the homodyne-technique). (d) As Fig. lc, but in dependence on co' and k.

scribed solely in terms of the M W D and the diffusion coefficient for c = 0. Since Eqn (8) does not contain the scattering-function P(k, M), it can be applied to particles of any arbitrary shape and size (e.g. linear coils in good solvents). F o r the determination of the M W D J(M) via Eqn (8), only the molecular weight dependence of the diffusion coefficient must be known. Equation (8) will be referred to in the following discussion as the basic equation for autocorrelation functions. The difference between the transition from Eqn (3) to Eqn (5) and the transition from Eqn (7) to Eqn (8) may be illustrated by one simple example. For this purpose assume that the scattering particles are monodisperse, i.e. the M W D is the delta-function * It will be shown later that such a case is in principle possible [cf. Eqn (1 la) and Eqns (35) and (38al],

f ( M ) = 6(M - Mo). Moreover, let the particle size be sufficiently small that P(k, M 0) ~- 1 for the observed scattering parameters. Finally, let As and B~ be two given constants.* Then the right-hand-side of Eqn (3), as a function of t, is described by the exponential function As.exp[-nk2D(Mo)t} + B~, the half-width of which (t~/2 = ln2/{nk2D(mo)}) tends to infinity for k--~0, in full accord with Eqn (5). In contrast the A C F (7}~which reduces to the form A s ' e x p { - n D ( M o ) t ' } + B, d o e s not depend on k and in the given example therefore is identical with the A C F (8) extrapolated to k = 0 ; the half-width (t'l,,z = ln2/{nD(Mo)}) is independent of k. The relationships are shown in Fig. I a,b for 2~0 = 5.145"10 Scm, n B = 1.5 and 0 _<0-< 150'; furthermore the diffusion coefficient D(Mo) = 5 10- 7 [cmZsec 1] was used (which corresponds, for example, to a mol. wt Mo ~- 7.4- 104 assuming Gaussian coils). The coefficients were assigned A~(k. c) -= 1

J. RACZEK

850

and B~(k, c) = 0, so that the systematised ACF C~ was in each case identical with the corresponding signalterm C~. It should be noted that the relationships do not basically change if other values for 2o, na, 0 and D(Mo) are chosen. The limits c---, 0 and k ~ 0 in Eqn (8) must in practice be replaced by extrapolations to c = 0 and k = 0. However, it is possible that such extrapolations may be easier to carry out in the coordinates k 2 and x/c than in k and c. Thus in the following, as a generalisation all ACFs will be regarded in dependence on the generalised coordinates g and h, and not on k and c. The generalised coordinate 9 represents a strictly increasing and smooth function of k, which vanishes for k - - , 0 ; then the inverse function g - a with the property g a(g(k)) = k exists, so that the dependence of the ACF on the scattering parameter can be eliminated by 9(k) ~ 0 instead of k ~ 0. The coordinate h is a function of c with the same properties as g(k), so that the concentration dependence can be eliminated by h(c)---~ 0 instead of c---~ 0. Use of g(k) and h(c) allows the basic Eqn (8) to be written in the form C(t') = lim C~(t', g(k), h(c)) h(c)~O O(k)~O

+ BA0,0). (8a) All other relations can be similarly rewritten.* Moreover, to simplify the following discussion, the time t will be formally replaced by the generalised time t', whereby the form of these equations does not change [e.g. in Eqns (1) and (2)]. EXTRAPOLATION-PROCEDURES

It is still to be shown how the systematised ACF (2) can be calculated from the corresponding measured ACF (1). In this process the following three conditions must be met: (1) The ACF C~ must be independent of the procedural and instrumental parameters (.. ,). (2) The signal-term Ci must remain unchanged. (3) In order to carry out the extrapolations h(c)---*,O and g(k)--~0, necessary for the determination of the MWD, the curves C~ for different concentrations and scattering parameters must be comparable; i.e. A~ and Bs must be smooth functions of h(c) and if(k) and furthermore A~(0,0)=~ 0 so that the information on the scattering system is not lost in the limit (8a). These conditions are certainly fulfilled when each single systematised ACF C~ for two given distinct generalised times t'l and t~ (with t'l < t[) assumes the required, given function-values C~1: = C~(t'l, g(k), h(c)) C~2 : = Cdt'2, g(k), h(c)),

(9)

which may depend on the scattering parameter via * Although the dependences of the ACFs on the coordinates t', (4 and h can differ from those on t, k and c, the symbols used up till now (e.g. C, P, As, A etc.) will be retained, since the continued use cannot lead to misunderstanding in the present work.

o(k) and on the concentration via h(c). The experimental section of this work and subsequent discussions 1-22] elucidate the optimal choice of the functions Cs~ and C~2. The corresponding experimental curves C~x take on the measured values

C~xx: = C¢~(t'1, g(k), h(c) . . . . ) C¢~2: = C,,(t'2, g(k), h(c) . . . . )

(10)

at the generalised times t't and t~. In order to calculate the ACF C~(t', g(k), h(c)) from the experimental curve, the coefficients A,(g(k), h(c)) and B,(g(k), h(c)) of Eqn (2) must be expressed in terms of the corresponding measured values C,x~ and C~,2 [see Eqns (1), (10)] and of the function-values C~1 and Cs2 required by Eqn (9); this results in the practical relation C~(t',g(k), h(c)) -

C~ - C~2 C~(t',g(k),h(c) . . . . ) C~,1 - C ~ 2

-

C~,2Csl - Ce, t C~2

(ll)

Cexl -- Cex2 which represents an affine transformation of Cox to Cs. Various systematised ACFs (11) for a set of data from one sample may now be easily compared with respect to h(c), g(k) and t'. The coefficients A~ and B~ defined by Eqn (11) are given by the expressions Csl - Cs2 AAg(k), h(c)) = - C , - Ci2 (1 la) B~(O(k), h(c)) -

Cs2 Cil -- Csl Ci2

Cil - Ci2

where Cil: = Ci(t'l, g(k), h(c)) and Ci2 : = Ci(t'2, g(k), h(c)) are the values of the signal-term at t'~ and t~ respectively. It is obvious from Eqn (l la) and Eqn (2) that the systematised ACF (ll) is--as required--independent of the procedural and instrumental parameters (...). It is furthermore clear that the coefficients As and B~ may be only indirectly determined if t'l 4= 0 and t~ 4= oo, since these coefficients are also dependent on the values of the signal-term at these time-points; i.e. the ACF (11) is systematised without the explicit knowledge of A~ and Bs. It also follows from Eqns (11) and (lla) that for all y(k) and h(c) the functions C~1 and C~2 must be bounded and must possess the property Another method for the determination of the systematised ACFs Cs is shown in the appendix to the present work. Two procedures will now be developed for the extrapolation of the systematised ACFs Cs to h(c) = 0 and subsequently to g(k) = 0: In the "first extrapolation-procedure", the generalised time t' is treated as parameter and for a given generalised time t' the limit h(c)---~ 0 and g(k)--+ 0 of all corresponding A C F - a m p l i t u d e s is determined. The procedure can thus be described by the relation C(t' = ~b)= lim Cs(t' = O,,q(k),h(c)),

(12)

h(c) ~ 0 o(k)~O

where ~b is an arbitrary, given constant. The extrapolation-step (12) thus yields the amplitude of the desired curve C(t') at t ' = 4~, i.e. a single point of

851

Determination of molecular weight distributions C(t').* To obtain the total curve C(t'), the extrapolation-step (12i must be repeated for a sufficient number of different q%values: in other words, the limiting curve will be constructed pointwise. In a graphical representation of one extrapolation-step (12) the wdues Cdt' = ch, g(k), h(c)) are plotted vs h(c) and/or ,q(k). In practice every measured A C F is given by N equidistant time-points t separated by the sampletime At = At(k). In order to make use of the full f-range of each individual A C F C~ for the determination of C(t') according to Eqn (12), the generalised sample-time At': = k 2.At - const

(13)

is introduced. This condition requires that all measured ACEs span the identical f-range NAt' = N k 2 A t for all concentrations c and all scattering parameters k. Thus each individual experimental curve must be recorded using such values of k and At that the condition (13) is satisfied. The constant value of At' used should be large enough that the essential part of the desired A C F C(t') is obtained by the extrapolation-procedure; this corresponds to a 3-4 fold half-width of C(t'), depending on the polydispersity of the scattering particles (see Eqn (32)). In the "second extrapolation-procedure", the amplitude of the systematised A C F is treated as parameter, and the limit t'~x. of the corresponding generalised times t' is determined via the extrapolations to h(c) = 0 and 0(k) = 0. Thus t' is expressed as a function of ,q(k) and h(c) and the double limit t~,~tr:

=

lim t'(g(kl, h(c):(,)

(14)

h(O ~ 0 y(k}~O

is determined, whereby ~ is defined by (.: = C~(t',g(k),h(c)) = const.

(14a)

For this purpose the values of t' used in Eqn (14) must be first determined by means of relation (14a). The extrapolation-step (14) yields that f-value which corresponds to C(t') = ~'.t To determine the total desired curve C(t'), the extrapolation-step (14) must once more be carried out for a sufficient number of different values of ~; in other words, here the extrapolated curve is also constructed pointwise. In a graphical representation of one extrapolation-step the values t'(~t(k), h(c); ~) are plotted vs h(c) and/or g(k). For the second extrapolation-procedure, it is not possible to give an explicit expression analogous to Eqn (13); this would a priori make use of properties of the scattering system which actually are to be determined by the experiment. It is nevertheless here useful to bear the condition 113) in mind, as will be demonstrated later. * Figure lb illustrates inter alia the limit ,q(k)= k---,0 in Eqn (12) for 4~ = t'l,2(k) =const. The value C(~) in this example agrees with all corresponding values Cs(t'1,2, k, C = 0 ) for k > 0. "I"Figure lb also illustrates the limit g(k) = k - - , 0 in Eqn (14) for the case C~(t',k,c = 0 ) - ~ = 1/2. The f-values obtained via Eqn (14a) are (in this example) the half-width values t'~ 2 fidentical for all k > 0) which agree with the desired t~.~..

At this point, the goal of the present work is achieved. By means of two different procedures an ACF can be constructed from the experimental DLScurves which is described by the basic Eqn (8a) [and (8)] and which is suitable for the determination of the MWD. To minimise the work involved, the analysis of the experimental curves (including the extrapolation-procedures) should be carried out numerically and a graphical representation should be used only to illustrate the relationships involved. To study the inverse problem i.e. the dependence of the distribution f ( M ) on the ACF C(t'k-it is assumed that the molecular weight dependence of the diffusion coefficient at vanishing concentration is given by D(M) = K D M ,,o

ao, KD > 0.

(15)

The substitution of the variable y: = D(MI = K o M -"'~ into Eqn (8a) then leads to the Laplacetransform [C~(t')] 1/" =

;/

F(_v)e " " d y = ~ [ F ( y t : t'],

(16)

where Ci(t') represents the signal-term of C(t') and the original F(y) is given by F(v)

la m ~-1/C2/ ...... I,~o+2)/,,.,rf~ K ,, 1)1 ..... }. (16a)

From Eqns (16) and (16a), it follows that the M W D is described by the relation f(M)=aDM.,KD

M .... 2 ~,6,- 1 t,[G(tt]' , 1,~ :y}.,

(171

which states that the inverse Laplace-transform of [C~(t')] t/" occurs in the expression for the MWD. Note that this conclusion is also valid when the D(M)relation is not described by Eqn (15). D is merely required to be a one-to-one function of M, which is practically always fulfilled: in this case, however, the Eqns (16a) and (17) must be modified accordingly. FREQUENCY SPECTRA It will now be briefly shown how the above developed procedures may be transferred to frequency spectra (FS). Experimental FS Sex can always be described as for the ACEs (1)- by the relation of the form Sex(U), k, c , . . . ) -- A'(k, c , . . . )" Si(¢o, k, c) + B'(k,c,...),

(lg)

where co is the frequency of the detector recording signal. In analogy to the ACFs, the signal-term Si, which is in general not describable by a theoretical expression, contains the total information on the scattering system. The coefficients A' and B' formally correspond to the coefficients A and B of Eqn (1). In analogy to the transition from Eqn (1) to Eqn (2), each experimental FS can be recalculated to a systematised frequency spectrum S, of the form Ss(~J,k,c) = A'Ak, c ) ' & ( ~ , k , c ) + B'~(k.c) without altering the signal-term.

(19)

852

J. RACZEK

Theoretical expressions for the signal-term are available for the case when the correlation of distinct particles vanishes (i.e. c ~ 0) and the scattering parameter k is sufficiently small. Under these conditions, the relation (19) for the heterodyne- and homodynetechnique leads to the integral-transform* lim Sh'(to, k , c ) = A~(k,0).0zMw) -1

c~O

~0 ~

D(M) (24a)

x ¢0,2 + D2(M ) d M + B~(0,0)

fo ° M f ( M ) P ( k , M )

lim S~m(to, k,c) = A'~(k,O).(zcM.,) 2

Mf(M)

= A~(O,O)(nM~)- '

o,12 ~ k2D(M) {k2D(M)} 2 dM + B'~(k,O)

(20a)

MM'f(M)f(M')P(k,M)P(k,M')

¢~0

×

k2D(M) + k2D(M ') dM' d M + B'~(k, 0). o~2 + Ik2D(M) + k2D(M')} 2

In the limit k ~ 0 the signal-term of the FS (20a,b) degenerates to the delta-function

S~(to, k, c) - B~(k, c) = 6(to), ~o A'~(k, c)

lim Si(to, k,c) = lira

~o

k~O

(22)

i.e. ~o' is the frequency to in units of k 2. Thus to' assumes also for k---, 0 all values to' > 0, provided that the frequency to tends to 0 as to oc k 2 (cf. Eqns (28) and (32a)). On substitution of to' into the Eqns (20a,b) it must be noted that the character of the spectral-density remains unchanged; i.e. for the Lorentzian in the FS of the heterodyne-technique, the relation

f

kZD(M)

to2 + {k2D(M)}2 d t o =

f

D(M)

~,2 + D2(M) dto'

(23)

must hold for k > 0, and an analogous equation for the Lorentzian in the FS of the homodyne-technique must also hold. I f - - a s in the treatment of the A C F s - the generalised coordinates h(c) and g(k) are introduced, the limit h(c) ~ 0 and 9(k) --, 0 of the FSs in dependence on to' is then given by Sht(to'): =

MM'f(M)f(M')

= A~(0,0). (rtMw) -2

since the Lorentzians degenerate to the deltaf u n c t i o n t This means that the limit k - ~ 0 leads to the total loss of the information contained in the FSs (20a,b), provided to is considered to be independent of k. To avoid this situation, a generalised frequency must be introduced (analogously to t') via the relation to': = ~ ,

lim S~m(to',g(k), h(c))

h(c)~O g(k)~O

(21)

k--O

to

shin(to'): =

(20b)

lim S~'(¢o',g(k), h(c)) h(c)~O o(k)~O

* In the present work, the symbol C for the autocorrelation functions relating to one recording technique will be indexed with "ht" for the heterodyne- and "hm" for the homodyne-technique respectively. Relations which are valid for both recording techniques will not be indexed. The frequency spectra (symbol S) are treated analogously. t The result (21) was expected to take this form, since the transition from the ACF to the corresponding FS is accomplished by a Fourier-transformation and the ACF (5) is a constant.

D(M) + D(M')

dM' dM

× to,2 + {D(M) + D(M')} 2 + B;(0, 0).

(24b)

The Eqns (24a,b) will be referred to as basic equations for frequency spectra. These depend on the M W D f ( M ) and the diffusion coefficient D(M) alone; the effects of concentration, intramolecular interference (scattering-function, intramolecular motion) and the procedural and instrumental parameters have all been eliminated from these expressions. Figure lc,d show the frequency spectra of monodisperse samples in dependence on co and to' respectively, whereby the identical assumptions were chosen as for the Fig. la,b; since A's = const and B's = 0, the systematised FSs are proportional to the corresponding signal-terms. It is clear from Fig. lc that the halfwidth ml/2 = nk2D(Mo) in the frequency-range tends to 0 for k--* 0; this is expected according to Eqn (21). On the other hand, Fig. ld shows that the half-width ell~ 2 = nD(Mo) in the generalised frequency-range is independent of k, since the FSs are identical for all values k >_ 0. Attention is drawn to the fact that the systematised FS Ss in Eqn (19) can be calculated from the measured FS (18) in a manner analogous to that described for C~ from Cex. The Eqns (9)-(11a) retain their validity for the frequency spectra, if t' is replaced by to' and the symbol C by S. Thus, for example, Eqn (1 l) modifies to the relation

S~(¢o', g(k ), h(c)) -

Ssl

-

-

Ss2

Sex I -- Sex 2

S~,(¢o', o(k ), h(c).... )

Sex2Ss~- S~xtSs2

(25)

Sex I - Sex 2

whereby S~1 and Sex2 represent the experimental values at the given generalised frequencies u~'l and co'2, while Ssl and

853

Determination of molecular weight distributions [S~(td',g(k),h(c),...)~

~.~-1 : convolution

S;~((d',g(k), h(c),...) ~

square

"

"~

chrn(t',g(k),h(c), ".-)

~ - root

~

C~(t',g(k),h(c),...)

Fig. 2. Schematic representation of the complete inverse problem..N, ~ . ~e denote respectively the Fourier-, Laplace- and Stieltjes-transformation. The symbol d' represents the extrapolation-procedures developed in the present work.

S~2 are the values which the systematised FS S~ must

F r o m Eqns (29) and (29a), the relation for the M W D

assume at lhese ¢o'~ and a)~. The extrapolation-procedures analogous to (12) and (14) are described by

J'(M) = 2aoKo~zMwM -'°- 2 ~ I ISht(U)'): ~] (30)

S(,/ = ~) = lim S~(c,/ = p #(k} h(c)) h(c)~O g(k) *0

with the constant

parameter

p

(26)

(first extrapolation-

procedure) and c')'~.: =

lim ~,/(#(kLh(c):a) hl,l~0

(27)

g(k)~O

with the parameter o-: = S~(~/,.q(k)h(c))= const (second extrapolation-procedure). In order to make use of the full e/-range of each individual FS S, by the practical extrapolation, the condition AU)

&o': = ~

~ const

(28)

i

then results. Equation (30) states that the inverse Stieltjes-transform of S)'(e) ') always occurs in the expression for the M W D . This statement is of general validity and is not limited to the D(M)-relation (15). The complete inverse problem of the DLS, describing the different possibilities for the determination of the M W D f ( M ) from experimental curves, is shown schematically in Fig. 2. The Fourier-transformation is depicted by the symbol f and the extrapolation-procedures (which presuppose the calculation of the systematised curves) are represented by the symbol & The inverse problem has already been briefly discussed [27], albeit purely formally. TESTING

with a suitable chosen constant should be observed: Ae~ = A.J(k) is the interval between two adjacent experimental points in the e)-range of a curve concerned and A./ is the corresponding interwd of the generalised frequencies. In studies of the inverse problem for the FS of the heterodyne-technique, it is necessar2¢ to introduce the variablc x: = D2(M). This leads to the Stiel(jestransform Sht(n),) =

~0~ (U'2G(X)+X dx = Yf[G(x);a/2},

(29)

in which sh'(~,/) is the signal-term of the FS (24a). For thc D(M)-relation (15) the original G(x) takes the form

G(x) =

1

1

(K~x 11',,,,'1) J t(KDX2

I)I/(2aD)I.

~Mw 2aD \,,'x (2%)

* In the present work, the symbol C for the autocorrelation functions relating to one recording technique will be indexed with "ht'" for the heterodyne- and "hm" for the homodyne-technique respectively. Relations which are valid for both recording techniques will not be indexed. The frequency spectra (symbol S) are treated analogously.

OF

THE

EXTRAPOLATION-PROCEDURES

The polystyrene TSK F-1600 of the Toyo-SodaManufact. (Tokyo) was used to check b o t h the extrapolation-procedures (12) and (14). The choice of toluene at 20~C as solvent ensured that the concentration effects had a strong influence on the experimental curves. Furthermore, the effects of the intramolecular interferences on these curves was considerable, due to the high molecular weight of the coils (M,, = 18.8.106, M,,/M, = 1.16 [18]) and their large expansion in toluene. Testing of the extrapolation-procedures was not carried out by determination of the M W D , but rather by evaluation of the z-average of the diffusion coefficient, since D_, is determinable directly without prior knowledge of any characteristics of the scattering system. In contrast, a determination of the M W D requires the D(M)-relation for monodisperse samples to be k n o w n ; a false result in the determination of the M W D would therefore be no real indication that the extrapolation-procedures are at fault. The autocorrelation functions Cho7' were recorded by the homodyne-technique using a 96-point-correlator from Malvern.* The spectrometer, which will be described elsewhere, was exactly adjusted in the range of angles 5 r _< 0 _< 155L The angle 0 could be set with an accuracy of A0 -~ 1/120 '~. For the primary lightsource, )-o = 5.145" 10 s cm (in vacuo, Ar+-laser). At this wavelength the refractive index for toluene is n~ = 1.5048.

J. RACZEK

854

Five concentrations c / ( j = 1. . . . . 5) of the sample were used. The ACFs were recorded at eight angles 0i (i = 1. . . . . 8) in the range 15°-50 ° at nearly equidistant intervals of 5 °. These angles and the correspondin9 sample-times Aq were calculated according to condition (13) using a constant value At' which was set (for this sample) to

/ 4rcnn'~2

"

20i

At' = k - ~ - 0 ) sm ~'Ati - 4.288-10 s Is c m - 2]. (31) For example, it follows from condition (31) that A t , ~ = 4 . 7 5 . 1 0 - S s must be associated with 0 4 = 29.96 °, while 07 --- 44.65 ° requires Atv = 2.20.10 -5 s. There are two possible ways of arriving at the constant for At' in Eqn (31). Firstly, if the value D(Mw) for the diffusion coefficient of the monodisperse sample with M: = M~ is known, the relation*

Figure 3 shows a measured A C F C~m by circles (O) in the scale of the right-hand y-axis; for the sake of clarity only each second point is plotted. The baseline of this curve is normalised to 1; this applies also for all other ACFs. In all figures the concentrations are given in units [g/(103 cm3)]. The experimental ACFs were smoothed by splinepolynomials in order largely to eliminate the statistical errors; among the published procedures, that of deBoor et al. [28] was particularly suitable. The values of these continuous spline-curves were used exclusively for Ch7 in the further treatment, rather than the experimental data themselves. The smoothed curves (for all values of cj and 0i) were then recalculated using Eqn (11) to the corresponding systematised ACFs C~m, in such a way that these always assumed the values c~i" chs7

At'

=

In 2 v n' D(Mw) N

(32)

can be used, in which the value of n = 2 applies for the homodyne-technique. The parameter v indicates how many half-widths the extrapolated ACF C(t') should roughly encompass (in the present case v = 2.5 was chosen). N is the number of the measured points of any individual ACF (N = 96 for the correlator used). D for monodisperse polystyrenes in toluene can, for example, be obtained [18]. Secondly, if the value for D ( . ~ ) is unknown, the constant At' can be estimated via preliminary experiments at the lowest concentration of the experimental series, namely from the half-width of an experimental ACF C~ at a suffÉciently small scattering parameter.

* The Eqn (32) results from an estimation of the halfwidth of the ACF C(t'). The analogous relation for frequency spectra is given by A~o'

= n' D(M,,) ~v.

(32a)

=

=

Cs~ " (tt,' g(k),h(c)) - 2 , C~hm(t2,

for t't = 6-At'

9(k), h(c)) - 1 for t~---* ~

Firstly, the desired A C F chm(t ') will be constructed from the systematised ACFs by the first extrapola-

c; m c°~

=

222° \

6

_o

20

40

(33)

according to Eqn (9). Thus C~'~ and Ch~' were chosen here independent of the concentration and scattering parameter. For the example in Fig. 3, C]" is shown in the scale of the left-hand y-axis. The values chosen for t'~ and t~ were the result of the following considerations: Errors in C~ml and C~72 give a falsely positioned systematised ACF C~" on application of Eqn (11); in other words, the coefficients A~ and B~ of C~" are not correct (see Eqn (lla)). This error-effect is minimised when t'~ and t~ lie as far apart as possible. On the other hand, t'~ and t'z must possess values where C~t and hm2 are recorded with the smallest possible error. This C,x is--inter alia--one important reason why the smoothed ACFs were used instead of the experimental values; in this manner the statistical error in Ch~'~is largely eliminated. In principle, the values of t'~ and t~ can also be chosen outside the measured range (e.g. t~ --~ 3c). Values t'~ < 3" At' should not be used since distortion due to the smoothing process is more probable at the two ends of the smoothed curves than in the middle.

0o,o,

60

eo

tTet'

Fig. 3. One experimental ACF C~x ~ and the corresponding systematised ACF C]" Efor the fixation (33)].

855

Determination of molecular weight distributions t i o n - p r o c e d u r e . T h o s e C~"-values w h i c h c o r r e s p o n d to a given generalised time t' = q~ were e x t r a p o l a t e d to h(c) = 0 a n d s u b s e q u e n t l y to g(k) = 0 a c c o r d i n g to E q n (12). T h e d i a g r a m s 4a and 4b illustrate h o w this pointwise e x t r a p o l a t i o n can be carried out. T h e m e a s u r e d Cf"-values (circles ©) are here p l o t t e d for t' = q5 = 25' At' a n d t' = 0 = 77" At' vs g(k) + h(c) = irk 2 + vc with suitable constants/,~ a n d v. As required, the ACF-values were first extrapolated to h(c) = vc = 0 for each #(kl) = ltk2: in this case this could be excellently carried out by a (numerical) linear regression in h(c). In Fig, 4a.b the results are s h o w n by crosses ( x ) . T h e e x t r a p o l a t i o n o f concentration was carried out for a sufficiently large n u m b e r of different t' = 0. thus yielding the total A C F C~" for h(c) = 0 at the g(ki) = l,ki e (i.e. Oi) u n d e r consideration. Figure 4c s h o w s for e x a m p l e the total A C F Ch," for the angle 02 = 19.56: e x t r a p o l a t e d to h(c)oc c = O, p l o t t e d vs the generalised time:* the (measured) systematised A C F s for the c o n c e n t r a t i o n s c5 a n d c3 are s h o w n only for c o m p a r i s o n . Similarly, Fig. 4d s h o w s the e x t r a p o l a t e d A C F CJ~" for 0s = 49.66' t o g e t h e r with the A C F at the c o n c e n t r a t i o n c5 ; all four Caj" for the c o n c e n t r a t i o n s c 1 to c4 for this angle lie b e t w e e n the two curves shown. H o w e v e r , the o r d e r i n g of the c o n c e n t r a t i o n s is here reversed c o m p a r e d to Fig. 4c; this fact can also be d e d u c e d from the e x t r a p o l a t i o n d i a g r a m s 4a,b. T h e vertical b r o k e n lines in Fig. 4c,d at the p o s i t i o n s t' = 25. At' a n d t' = 77. zXt' serve only to facilitate the c o m p a r i s o n with the d i a g r a m s 4a,b. The e x t r a p o l a t i o n g ( k ) = l,k2--+O o f the A C F values, already e x t r a p o l a t e d to h(c) = 0, could n o w be carried out for each t' = 0. This yields in each case the value C a m ( f = 4)). F o r this purpose, the A C F values for h(cl -- 0 were fitted numerically by a q u a d ratic p o l y n o m i a l in g(k) 3c kZ; the e x t r a p o l a t e d point * The total ACFs, extrapolated to h(c)~O (and ,q(k)-+ 0). can be regarded as quasi-continuous functions of t', since a sufficiently large number of different q~-values was used in practice: thus these ACFs in all figures are shown as continuous curves.

is then given by the p o l y n o m i a l at ,q(k) = 0. This process was suggested by c o n s i d e r a t i o n o f the d i a g r a m s 4a,b, w h e r e the e x t r a p o l a t e d value Chm(t ' = ~b) is represented by the ),-intercept of the curve for h(c) = 0. T h e desired total A C F Ch"(t'), w h i c h is described by the basic E q n (Sa), is o b t a i n e d in repetition o f the e x t r a p o l a t i o n g(k)---+ 0 for a sufficiently large n u m b e r o f different qS-values. This A C F is s h o w n in Fig. 4e t o g e t h e r with several o t h e r A C F s e x t r a p o l a t e d to h(c) = 0 at finite angles 0i.* Figure 4a,b and all following diagrams show the numerically determined extrapolation-curves which describe the extrapolation-sequence used: first h(c)---+0 and then g(k) -+ O. To give an impression of the total structure of the diagrams, the extrapolation-curves describing g(k) ~ 0 at the given concentrations c / a n d the extrapolation-curve describing h(c)--+ 0 at g ( k ) = 0 are also shown. The latter curves illustrate in principle the reverse order of the extrapolation-sequence (i.e. first g(k)---+ 0, then h(c)-+ 0), and were determined merely by optical/graphical means. It has been shown [22] that this sequence of extrapolations is more difficult to perform numerically than the above-mentioned. A future publication will deal with the problem of which conditions must be observed so that this latter sequence of extrapolations may be used at all for the practical determination of Ch"(t ' = 4)). In order to illustrate the relationships involved in the first extrapolation-procedure for other C]'~ and C]~', these were set

C~'~ = C]"(t'l,g(k),h(c})=

10 1.0 + 104.c - 15 for t'~ = 3. At'

In contrast to the earlier fixation (33) of the ACFs, C~]' is chosen here to be dependent on both concentration and angle. For the purposes of the recalculation of C~m according to Eqn (11), only the smoothed ACFs were used, as before. The extrapolations for the determinaton of C""{t') were carried out as described above. Figure 5a shows the extrapolation-diagram for t' = 4~ = 25. At'; this possesses a form quite different from those of Fig. 4a,b. One marked difference lies in the fact that for 0 < 35' the concentration

ch'~(25 a', i.l.k2,v c) ~

1.72

S

j. J

~

0 ~o °

~

O, =29,96°

\ g(k),h(c) .

0

.

.

.

.

.

.

,

20

10

Fig. 4(a) I!.P1181(I

I~

(34)

C~7 = C~"(t'2, g(k}, h(c)) = 0 for t~ = 80. At'.

30

ILk 2 • v e

J, RACZEK

856 ,

i

C=hm(7?At;pk2,vc) 1.28

~--0" ..~:14,s5° ~

.

.,~=19,56°

~

~o~4.8o.

~

~=39,72 °

I'081c-~o o

•

20

lo

3'0

g(k)+h(¢) ILk2+vc

Fig, 4(b~

cshm(~.',p,k2,VC)

2£

~

/

~2=19,56"

c--O/ c3=0'2089 ~ ~ ~ ¢5=0'3511

I

60

40

20

80

t'/et'

Fig. 4(c)

c,~'(t'.~k'.vc) 08=49,66*

i~

o-o

i 1

20

- -

td)

60

Fig. 4(d)

~

t'lt~t'

Determination of molecular weight distributions

2.C~ ;

c--O I

i

857

i

,.S-O o

I

i

i

I

I0

o

L

40

20

60

8o

t'/&t'

Fig. 4(e) Fig. 4.(a). Extrapolation-diagram for C]" values [fixation (33)] of TSK F-1600 at the generalised time = 25.At', in dependence on y(k)+ h(c)= # k 2 + vc. 0 experimental values (smoothed). x values extrapolated numerically to h(c) = 0. (b) As Fig. 4a, but for ~b = 77. At'. (c) The ACFs C~" [fixation (33)] for TSK F-1600 at 02 = 19.56 and c 5, c s shown together with the corresponding to h(c) ~c c = 0 extrapolated curve, in dependence on t'. The broken vertical lines are to facilitate a comparison with the Figs 4a,b. (d) As Fig. 4c, but for 0s = 49.6& (the ACF for c~ is not shown). (e) ACFs C~" (extrapolated to h(c) = 0) for TSK F-t600 at various angles 0~ > 0' shown together with the corresponding ACF for 0 = 0 , in dependence on t' [fixation (33)]. The vertical lines indicate the first extrapolation-procedure, the horizontal lines the second-procedure. dependence of the ACF-values is non-linear. Consequently all numerical extrapolations were carried out using polynomials of the first and second degree in h(c) = vc and ,q(k) = pk 2. The ACFs of Fig. 5b and c correspond to those shown in Fig. 4c and e; it should be noted that the content of information on the scattering system of the corresponding ACFs is identical, since the respective signal-terms are identical. However, the distinct fixations (33) and (34) have the consequence that the individual ACFs Cam are quite d!~'erently weiyhted in the application of the extrapolationprocedures. Note furthermore that no use is made here of the fact that the baselines of the experimental ACFs are known; this is important for the use of several types of correlators.

Analysis o f the extrapolation-diagrams s h o w s that these are not necessarily unique; o n the o t h e r h a n d it is clear that the i n f o r m a t i o n a l c o n t e n t o n the scattering system is identical in each case. T h e form of the d i a g r a m s always d e p e n d s on three factors: (at the signal-term, which c o n t a i n s all the i n f o r m a t i o n on the scattering system, (b) the functions h(c) a n d ,q(k) a n d (c) the coefficients As a n d B~ used. In the present m e t h o d o f calculation o f C]", the coefficients As and Bs themselves are d e t e r m i n e d [ a c c o r d i n g to E q n (1 la)] by the signal-term a n d the functions C]~ a n d C]~' at the generalised times (1 a n d t~ (i.e. there are four degrees o f freedom). Therefore, the projective

C~(25At'.pk2.vc) -1

i]~_7_-f--J

--2 l

o

l--4 l

~

~

,-

u

~

II

~7=44.ss"

~______~-~j~

~

~

..~

"~4= 29,96" '~.~: 24.96"

~,:3,.ao.

~2=tg,s6"

~0"

0

10

20

Fig. 5(a)

30

Ilk 2 * v c

858

J. R A C Z E K

C~(t'.lak%c)

_5 ¸

-10

-15 o

zo

~o

so

so

t'/&t'

80

t'/nt'

Fig. 5(b)

C~(t'..k~.vc) o I

-5.0

7: 7//

_,o.o

o_27

o-0

-15.0 3

20

40

60

Fig. 5(c) Fig. 5. (a) Extrapolation-diagram for Ch/"-values [fixation (34)] of TSK F-1600 at the generalised time q~ = 25.At', in dependence on O(k) + h(c) = #k 2 + vc. 0 experimental values (smoothed). × values extrapolated numerically to h(c) = 0. (b) The ACFs C~" [fixation (34)3 for TSK F-1600 at 02 = 19.56 ° and c5, c3 shown together with the corresponding to h(c) occ = 0 extrapolated curve, in dependence on t'. The broken vertical line is to facilitate a comparison with Fig. 5a. (c) ACFs C~" (extrapolated to h(c) = 0) for TSK F-1600 at various angles 01 > 0 ° shown together with the corresponding ACF for 0 = 0 °, in dependence on t' [fixation (34)]. The broken vertical is to facilitate a comparison with Fig. 5a. diagrams used here do not furnish unique values of A~ and Bs and should not be referred to as Z i m m diagrams.*

* Projective diagrams, such as for example the diagrams of the present work, or Zimm-diagrams (CLS), are familiar in projective geometry. The importance of Zimm-diagrams lies in the relatively simple and accurate simultaneous determination of various quantities (~rw, ( r - ~ , A2, P(k,M)z) from CLS-measurements. The Berry-diagrams (square-root representation, also CLS) are a further example of projective diagrams.

The extrapolation-procedure was next checked by the calculation of the z-average of the diffusion coefficient, D~. It has been shown [18] that D z can be determined directly from the extrapolated A C F chm(t'); the use of this m e t h o d for both fixations (33) and (34) gave in each case the value D~ = 2.07" 10 - s [cm 2 s - l ] . Further, D~ was determined by a conventional procedure, in which the m e t h o d of cumulants was applied to each single experimental curve, yielding an apparent diffusion coefficient (Dz)kc for the scattering parameter and concentration concerned. The extrapolation of (Dz)kc to c = 0 and k = 0 resulted in the average D~ = (Dz)oo = 2.05' 10 a [cm 2 s - ~]. This

Determination of molecular weight distributions

859

t'(~kZ, vc; {; ) 26

~ --' C~(t',p.k~,vc)=13139

20]

.b-o" ,~1=1t.,85"

\ - ~

~

\

"~,

(3

~

o

~

~

~

10

+,=,,.s+.

~

20

~,o

,---~0 -~: 49.66"

30

g(k)+ h(c) I.~k2 + Vc

Fig. 6. Extrapolation-diagram for t'-values which correspond to the ACF-value ( = C~'(t', ltk 2, vc) = 1.7139 [second extrapolation-procedure; fixation (33)] of TSK F-1600, in dependence on g(k) + h(c) = l~k2 + vc. © experimental values (numerically calculated from the continuous spline-curves), x values extrapolated numerically to h(c) = 0. value agrees very well with that given above. This shows that the first extrapolation-procedure, even under the exceptionally critical testing-conditions of the example used, gives an acceptable accurate A C F Ch"(t'); one i m p o r t a n t precondition for the determination of the M W D from Ch"(t ') is thereby fulfilled. The second extrapolation-procedure was now carried out for the fixation (33). Because of E q n (14a), a fixed ACF-value ( = C~"(t',g(k),h(c)) was assigned: the corresponding F-values were determined numerically using the spline-curves. These F-values were then first extrapolated to h ( c ) = 0 at constant g(ki) by means of a linear regression; use of a quadratic polynomial in ,q(k) enabled the further extrapolation to g(k) = 0 in order to obtain the desired t'~x,,. The expressions #(k)=-Hk 2 and h ( c ) = w~ were applied as before. The extrapolation-step (14) is illustrated for (, = C]"(t', g(k), h(c)) = 1.7139 in Fig. 6; it is clear that the extrapolated value t'c~,~ is practically equal to 25' At'. This gives good agreement with the first extrapolation-procedure, which yielded the (-value given above for q5 = 25' At'. Both extrapolation-procedures are consequently performed with a high degree of accuracy. It should be noted that the form of the extrapolation-diagram (6) is similar to that of Fig. 4a. The extrapolation-step (14) was carried out for a sufficiently large n u m b e r of different (-values, yielding the total desired A C F ch"(t ') (see Fig. 4e). The second extrapolation-procedure possesses the

disadr'anta.qe that parts of some A C F s at b o t h ends of the measured t'-range may not always be usable. This can be simply illustrated by consideration of the A C F s in Fig. 4c,d and e at t' < 2. At'. O n the other hand, it has the advantage that the extrapolated r-values are not confined to the experimental Y-range; this is indicated by the circle (O) in Fig. 4e on the right-hand-side. Sufficiently accurate measure-

ments and extrapolations yield an extrapolated A C F C~"(t ') identical to that obtained by the first extrapolation-procedure, but in a different t'-range: the first procedure always yields the same t'-range as measured, but in the ,second procedure this is also dependent on the signal-term of the scattering system under consideration. It can accordingly be advantageous to combine the two extrapolation-procedures in such a way that the first is applied for the measured range At' < t' < NAt' and the second is used for the extremes of this range; this would give the A C F Ch'(t ') over a larger (-range than was actually measured. Finally, the following two remarks should be made: Firstly, the extrapolation-procedures here developed provide for the first time a method for the interpretation of the ACFs (of FSsJ with respect to intramolecular motions directly at infinite dilution, as is demanded by different theories (e.g. for Gaussian coils [29]). This involves the registration of the ACFs (or FSs) at a single suitable scattering parameter, and the subsequent extrapolation of the systematised curves to h(c)-+ 0, but not g(k)--+ 0: therefore, it is unimportant in this case whether t' or t (+,)' or +,)) are used as variables. Secondly, it arises in many other problems quite apart from dynamic light scattering that experimental curves at finite concentrations are available, but a theoretical expression is only known for c = 0. To allow the use of the theoretical expression, it is possible however in most cases (as for DLS) to construct a curve for c = 0 from a system of experimental curves. To this end, systematised curves can in general be calculated either by Eqn (11) or (often probably simpler) by Eqns (38) and (39), whereby t' and C must however be accordingly interpreted afresh. The above described extrapolation-procedures with respect to h(c) --, 0 can then subsequently be applied; the further limit g(k)-+ 0 has no meaning, when there exists no quantity analogous to k. For example, in ultracentrifuge measurements t' (DLS) must be replaced by the sedimentation coefficient s and C~, C+ as the (measured, systematised) distributions of

860

J. RACZEK

s; c is the mean concentration (as for DLS), but k must be replaced by lit since the limits 1/t--*O and c---*0 are required (t: centrifugation time). Furthermore, a modified extrapolation-procedure can be applied in the determination of the MWD via classical light scattering coupled with GPC-measurements, such as possible with the Chromatix assembly; in this case all concentration effects are eliminated which otherwise are not negligible and to date have caused many difficulties in their treatment.

Sciences Department CSD TR 20 and 21, Purdue University, West Lafayette, Indiana (1968); (b) C. de Boor, A Practical Guide to Splines, chapter XIV. SpringerVerlag, New York (1978); (c) IMSL-library (editions 6 to 8), prog.: ICSFKU (IMSL, Customer Relations, Sixth Floor, NBC Building, 7500 Bellaire Boulevard, Houston, Texas 77036, U.S.A.). 29. R. Pecora, J. chem. Phys. 40, 1604 (1964); 43, 1562 (1965); 49, 1032 (1968).

Acknowledgements--The author expresses his thanks to Professor G. Meyerhoff for numerous helpful discussions. Support of this work by the "Sonderforschungsbereich, Chemie und Physik der Makromolektile" is gratefully acknowledged. REFERENCES

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APPENDIX

The systematised ACFs Cs must possess the three properties given in section 3. However, these do not rigorously prescribe how the A C F s C~ must be calculated. One possibility for the calculation of Cs frord C~x is given by Eqns (9)-(11); this does not require a priori knowledge of A a n d / o r B in Eqn (1). F o r some types of correlator, however, the baseline B(9(k), h(c) . . . . ) of the experimental curves is known. This can be used for a second calculation-method of C, which will now be described. For this purpose it is assumed that B(y(k ), h(c) .... ) = B = const

and that all ACFs are recorded in the identical generalised time-range, i.e. according to Eqn (13) At' <_ t' <_ NAt' must hold in the experiment. Under these conditions, the systematised ACFs Cs can be calculated by fixin 9 their integrals, i.e. setting NAt'

l~(g(k), h(c)): = |

C~(t', 9(k), h(c)) dt'

f c~ dt'

(36)

t'

:

+ (N - l)at'B,(9(k), h(c)). I, is here a given, bounded and smooth function of g and h for which in the limits h(c) --~ 0 and g(k) --~ 0 the relation lira I~(g(k), h(c)) ~ (N - l)At'B~(O,O)

(36a)

h(c) ~ 0

o(k)~O

holds. For the experimental curves, the integral which corresponds to Eqn (36) is given by N A t'

lex(y(k),h(c) .... ): = |

Cex(t',y(k),h(c) .... )dt'

(37)

t' N At'

= A(o(k),h(c) .... ) |

d a t'

C,(t',g(k),h(c))dt' + (N - l)At'B.

If A from Eqn (37) is substituted into Eqn (l), and As from Eqn (36) likewise in Eqn (2) and if from the two resulting relations the term ~ Ci dr' is then eliminated, the systematised ACF C~(t',g(k), h(c)) =

* The coefficient A~ defined by Eqn (38), is given by Is - (N - 1)At'B~ As = (38a)

(35)

I~ - (N - I)At'B~ l¢x - (N - I)At'B

x {C~x(t',9(k),h(c) .... ) - B} + B,

(38)

Determination of molecular weight distributions is obtained.* In order to simplify the presentation of Eqn (38), the dependences of B~, I s, lex on 9(k), h(c) and (...) were omitted. To calculate C, using Eqn (38), the coefficient B~ must be given in addition to Is, whereby B s must be a bounded and smooth function of y and h. In practice, one will generally be able to choose a constant value for B~, e.g. Bs(g(k ), h(c)) ~ O. It may come about that use of the sums of ACF-values (rather than integrals) is more convenient for the calculation of the systematised ACFs; in this case the experimental curves need not be smoothed nor numerically integrated. Thus, if lex and Is are now taken to represent the sum of the experimental points of one ACF and the sum of the systematised ACF-values respectively, each at the corresponding t' = i. At' (i = 1,. , N), then the relation

861

I ~ - (N - l)B~ "~ Cs(t',g(k),h(c)) - I~,-- (-A; -- I)B .Cex(t"y[k)'h(c) .... ) -

B} +

Ss

(39)

results. T h e previously described e x t r a p o l a t i o n - p r o c e d u r e s (12) a n d (14) are n o w directly applicable to the A C F s w h i c h have b e e n calculated via E q n s (38) or (39). It s h o u l d also be n o t e d t h a t further calculationp r o c e d u r e s of C s exist b u t are n o t dealt with here. These a r e - - l i k e the p r o c e d u r e s (11), (38) a n d ( 3 9 ) ~ a l s o applicable to frequency spectra (and o t h e r e x p e r i m e n t a l methods).