Linear combination of two-parameter functions for the molecular-weight distribution of polymers

European Polymer Journal, 1973, Vol. 9, pp. 805-814. Pergamon Press. Printed in England.

LINEAR COMBINATION OF TWO-PARAMETER FUNCTIONS FOR THE MOLECULAR-WEIGHT DISTRIBUTION OF POLYMERS E. SAIZ and A. HORTA Departamento de Quimica Fisica, Facultad de Ciencias, Universidad Complutense, Madrid-3, Spain

(Received27 January 1973) Abstraet--A linear combination of the exponential, logarithmic and normal functions is proposed for the empirical description of molecular-weight distributions in fractionated polymers. This linear combination contains four adjustable parameters. Two of them are fixed by the experimental values of )17/,and )~w. The remaining parameters are determined by two different methods. One uses )ktz and ~tz+ 1, while the other looks for the least-squares fit of the fractionation data. A numerical example of the usefulness of this linear combination is given by applying both methods to a fractionation of poly-(methylacrylate). I. I N T R O D U C T I O N IT IS well known that molecular weight for polymers is not a single-valued quantity but, rather, has a statistical character. This has far-reaching consequences on the properties of polymeric materials and causes numerous difficulties in characterization. According to statistical laws, the maximum information that can be obtained about the molecular weight of a polymer is contained in its probability distribution. This distribution can be expressed in terms of the different bases for counting (number, weight, etc.), the most frequently used being the weight distribution. We shall refer to WM for this distribution and to M for the variable designating molecular weight. Wu is normalized in such a way that WMdM expresses the weight fraction of polymer having molecular weight in the interval between M and M + dM. The problem of finding the functional form of WM corresponding to a given polymer has been intensively studied. ~1) A simple representation of WM is possible if only two parameters are required to determine the distribution. Different forms of two-parameter functions (TPF) have been proposed, and their application to polymers has been extensively discussed3 I) The parameters chosen to determine these functions are usually the first two averages of M i.e. the number and weight averages )~r and For those polymeric systems in which experimental determination of more than just two averages of molecular weight is possible, a T P F does not provide enough flexibility to describe the distribution. For instance, the techniques of analytical fractionation of polymers (precipitation, extraction, GPC, etc.) yield very detailed information about the distribution of M values in a given sample and, in principle, allow a characterization of WM more accurate than that represented by a TPF. Additional parameters should then be introduced in the functional form of WM properly to describe fractionated polymers. The need for additional parameters has been recognized by several authors. Proposals for new parameters include threeparameter functions ~2,3) and infinite series expansions of WM in terms of orthogonal 8o5

806

E. SAIZ and A. HORTA

functions. (4) To be really useful, however, the introduction of new parameters should not increase the mathematical complexity of the manipulation of results. As an attempt towards this objective, we propose in this paper the use of a linear combination of several simple TPF's all having the same/fin and Mw. This introduces new adjustable parameters in the form of coefficients but obviates the use of any new mathematical expression, either for WM or for the averages of M because the linear combination merely adds functions which are already well known and easy to handle. As a starting point, we use a linear combination of three different TPF's. The number of adjustable parameters is then four, because the normalization condition for WM eliminates one of the coefficients. Two of these parameters are taken to be/fin and Mw so that these averages have the same value in the linear combination and in each of the individual TPF's. The other two parameters can be determined in different ways; in Section 3 we present two possible methods, one using -~z and/ff~ + 1 (the third and fourth averages of the polymer molecular weight) and the other looking for the least-squares fit of WM to the experimental data of a fractionation. For the linear combination to be of wide applicability, the three TPF's that form its basis should have very different behaviour so that the resulting WM could cover many different possibilities. The three functions chosen to this end are the exponential, logarithmic and normal functions. In Section 2 we give their definitions and emphasize their differences.

2. T W O - P A R A M E T E R F U N C T I O N S We express the weight probability distributions in terms of the reduced variable x defined as x - M/lff,. The relationship between Wx (the distribution in terms of the variable x) and WM is very easy to establish since the probability has to be conserved under a change of variables. That is:

Wx dx ---- WM dM,

(1)

which amounts to W~ = lff, Wu. The use of x instead of M is convenient because it allows a straightforward comparison of the different TPF's in terms of only one parameter, the polydispersity ratio r (defined as r - Mw//ff,), regardless of the size of the polymer. The definitions of the exponential, logarithmic and normal functjons, in terms of x and r, are as follows* :is)

(x~

1/~'-',

r W~ ---- ~

x

)~ exp{-- [ln(xr~)12/21nr} exp(½ In r)

WxN = [2~r(r -- 1)] -~r x exp[-- (x -- l)Z/2(r -- 1)].

(3)

(4)

Here, the symbol F(y) denotes gamma function. ¢6) * Superscripts, E, L, N, are used throughout this paper to indicate that the superscripted magnitude corresponds to the exponential, logarithmic or normal function, respectively.

Two-parameter Functions for Molecular Weight of Polymers

807

Two different characteristics of this set of three functions make them especially useful as a basis for the linear combination. The first is that their maximum value of Wx varies with r (along x) in a different way for each one of them. Let us call x.p and hTt.p the most probable values of x and M, respectively (Xmp = M . p / M . ) . The dependence of x.p on r is found simply by imposing the condition d W x / d x = 0 on Eqns. (2)-(4). The result is x.p e ---- 1

(5)

Xz; L = r - ~

(6)

XmpN = ½ + (r - - ~)*.

(7)

These variations are illustrated in Fig. 1. In the first case, x,,p is constant; in the second, it decreases with r, and in the third, increases. In terms of molecular weight, this means that, with increasing polydispersity, Mm~, gets increasingly smaller (larger) than 3~, in the case of the logarithmic (normal) function, while it is always coincident with 3~t, in 3 ---

I rithmic

2

Exponential

I Normal

I

I

I

Mmp f

Exponential L.ogarlthmic

I

I

I

2

3

4

Mw/Mn

FIG. 1. Most probable value (M.t) and z-average value (M.) of molecular weight as a function of polydispersity ratio, for the exponential, logarithmic, and normal distributions. the case of the exponential one. These three functions, thus, cover all possible cases with respect to the localization of the maximum in the molecular-weight distribution. The second characteristic, making this set of three functions useful, is that they

808

E. SAIZ and A. HORTA

represent a different influence of polydispersity on the averages of molecular weight. Let us call rk the k-th average of the reduced variable x, defined as rk ----

W~ x k- 1 dx

0

Wx x k- 2 dx.

(8)

0

Then, the rk's are functions of the polydispersity ratio only and are given by rkr = 2 - r + ( r - l ) k

(9)

rkz = r k-1

(10)

(~-f-)* r~N =

Hk{[2(1-- r)]-~'}

(11)

Hk_l {[2(1 -- r)]-*}'

where H~(y) stands for the k-th degree Hermite polynomial3 6) The averages of molecular weight are expressed in terms of rk as follows k = 2 : 2Q~ -----AT".r

(r = r 2 )

k = 3: 2Q'~ = ~

(r, ---- ra)

r,

k=4:2Q',+x=)VT.r,+t

(12)

(r~+t = r , ) .

The variation of rz/r with r [where rz is expressed by Eqns. (9)-(12)] is shown also on Fig. 1. We see that the logarithmic and normal functions represent positive and negative deviations respectively from the behaviour of the exponential function. This means that a wide spectrum of possible values of )fix is covered by the three distributions. The same holds true for averages of M higher than _~7~. 3. L I N E A R COMBINATION We write now the distribution function Wx, which represents a given polymer, as a linear combination of the three TPF's described in section 2: W~ (linear combination) = aWx ~ + /3W~L + yW~ N,

(13)

with WxE, Vexz, and W~Ngiven by Eqns. (2)-(4). The coefficients a,/3, )', in the combination are pure numbers obeying the normalization condition a = 1--/3--),.

(14)

Eqn. (13) expresses, thus, a distribution containing four parameters which are the following: 21~,, _~tw, /3 and )'. The experimental determination of 37, and .bIw, for a given polymer sample, yields the value of its polydispersity ratio r, which is to be plugged into Eqns. (2)-(4) for Wx r', Wx L, Wx N. M, also allows the transformation of W~ into WM as WM ----- Wx/M,. The two independent coefficients/3 and )', remaining as adjustable parameters, can be determined in many ways. They should give a linear combination which best represents the real distribution of the polymer sample. We propose here two different methods to calculate those coefficients, viz. method A and method B.

Two-parameter Functions for Molecular Weight of Polymers

809

Method A

Let us assume that the values of M, and klz + 1 can be determined experimentally as well as kSrn and Mw. Then, it is possible to find/3 and ~, such that the resulting theoretical distribution has its first four averages exactly coincident with the experimentally determined M~, J~w, ~ r and M~ + 1. This is achieved by writing the following condition of equality between experimental and theoretical averages: 37r~ -----ctM, n +/3M, ~- + yM, N

(15) (16)

M~+~ = a M , + t n + / 3 M , + , L + y M , + l N.

These two equations, together with the normalization condition [Eqn. (14)], yield the solution /3 -----[(M,+l -- Mz+l E) ( M , " -- Mz ~) -- ( M , - - M , ~) ( M , + , N -- M,+ln)]/A

(17) ~, =

[(m,+~'

-- e,,:)

(mz -- m:)

-- (m~ L -- m:)

(m,+l

-- m,+~)]/A

(18) A = [ ( M z + l L - - M z + ~ E) ( M z ~ - - M , E) - - ( M , L - - M~ E) ( M z + ~ N - - M ~ + t g ) ] .

(19) The theoretical z and z + 1 averages of the TPF's, which are here required to calculate /3 and ~, can be obtained from Eqns. (9)-(12) with the Mn and 3twvalues measured on the polymer. Method B

This second method can be applied when the distribution of molecular weights has been determined experimentally by fractionation. Then, it is possible to find those values of/3 and y which produce the least quadratic deviation between theoretical and experimental distributions. Let us call xi the value of the reduced molecular weight corresponding to the i-th fraction of a polymer. The mean quadratic deviation, A, between the experimental Wx and the theoretical linear combination is defined by = n -1 ~

(wx, - ,~W~,E -/3W~," - y W ~ " ) ~,

(20)

l=l

where n is the total number of fractions detected in the polymer. The conditions for a minimum values of A, with respect to/3 and y, are: ~9A/3/3 = cgA/~gy---- 0. These two conditions, applied to Eqn. (20) with a given by Eqn. (14), yield the solution

l

1

f

1

(21

810

E. SALE a n d A. H O R T A

l

1

i

(22/ l

l

--[~

(Wxf -- Wx,L) (W~,r -

W~,N)1.2

(23)

1

These values of fl and ~ represent the least-squares fit of the theoretical distribution Eqn. (13) to the experimental data for the fractionated polymer. Application to experimental data

The usefulness of the linear combination is now demonstrated by applying methods A and B to the experimental data of some fractionated polymers. The data are taken from L6pez Madruga et al. <7~ and correspond to three samples of poly-(methylacrylate) fractionated by precipitation at 25 ° using acetone as solvent and a mixture

TABLE I. MOLECULAR-WEIGHT AVERAGES CALCULATED FROM THE FRACTIONATION DATA REPORTED BY I.~PEZ MADRUGA et al. (7) FOR POLY-(METHYLACRYLATE)

Mavcr.se × 10 - 5

Sample identification

Number of fractions

/F/,

JF/w

-~/'z

jiT/~+ 1

R2 R1 R3

17 20 14

0"843 1.08 2"29

1.21 1.61 3"29

1.59 2.08 4.09

1.91 2"42 4.66

of methanol and water (2:1) as non-solvent. The molecular weight averages of those three samples, calculated from the fractionation data, are shown in Table 1. Equations (17)-(19) of method A can be applied directly to these averages and the resulting values of the coefficients appear in Table 2. TABLE 2. COEFFICIENTS OF THE LINEAR COMBINATION CALCULATEDACCORDING TO METHODSA AND B Method A

Method B

Sample identification

a

/3

~,

a

/~

~,

R2 R1 R3

1-99 1.38 1"47

--0"38 --0.32 --0"43

--0"61 --0-06 --0-04

2"13 1"81 2.72

--0"63 --0-57 -- 1.26

--0.50 --0.24 --0"46

T w o - p a r a m e t e r F u n c t i o n s for Molecular W e i g h t o f Polymers

811

The fractionation data cannot be used directly in Eqns. (21)-(23) of method B because the measured weight fractions are not the distribution itself. First, we convert these weight fractions w~ into cumulative values C~ by the simple method of Schulz ts) l--1

c, = ½ w~ + ~

(24)

w~.,

J=l

This cumulative experimental curve can be transformed into the distribution Wx~ by numerical differentiation. Instead of differentiating the experimental cumulative curve, we prefer to integrate the theoretical Wx. In this way, the C~ values can be used directly. The coefficients/3 and ~, of method B are then those which produce the least squares fit of the integrated linear combination to the experimental cumulative points Ct. Equations (21)-(23) are used as they stand for the calculation of 13 and y, simply replacing Wx~by C~ and the differential theoretical Wx,'sbytheircorresponding integral functions C~, defined by XI t~t

cx,"

= I dx Wx",

(25)

*/

o

where ~7is either E, L or N. These integrals have been carried out, for each measured fraction, up to its reduced molecular weight x,, using numerical quadrature (Simpson's rule). The coefficients thus obtained with method B are shown in Table 2. The mean quadratic deviation, ;~, for the cumulative curve can be calculated using Eqn. (20) if we replace Wxj by C~ and the W~7's by C~7. The values of)t thus calculated for the linear combinations obtained with method A and with method B are shown in Table 3. Also shown for comparison are the mean quadratic deviations which are obtained when the cumulative points are fitted to each individual TPF with the same h~', and )ff,~ as the linear combination. TABLE 3. MEAN QUADRATIC DEVIATION (,~) BETWEEN THEORETICAL INTEGRAL CURVES AND EXPERIMENTAL CUMULATIVEPOINTS OF THE FRACTIONATION

A × 10 a Sample identification R2 R1 R3

Method A

Method B

Exponential

Logarithmic

Normal

0.50 0.79 0.50

0.36 0.70 0.19

0.96 0.98 0.92

1.92 2.69 3.37

6.72 5.45 3.17

The values of M'z and )t~ + l, calculated for the linear combination obtained with method B and for each individual TPF having the same )1~, and/ffw, are given in Table 4. 4. DISCUSSION The advantages of the linear combination of TPF's for the polymer molecularweight distribution are clearly shown by the results contained in Tables 3 and 4. The combination is, in fact, superior to any of the single TPF's when applied to the

R2 R1 R3

Sample identification

1-53 2"07 3"99

1.75 2-30 3"97

1-52 2-14 4-24

1"86 2-66 5.21

Exponential Mz Mz + i 1.66 2"39 4-65

2.33 3"56 6.59

Logarithmic Mffi Mz + i

Mav©ra, e X 1 0 - s

1"33 1"79 3"67

1"49 2.04 4"13

Normal Mz Mz + i

0.748 1.03 2.16

1"27 1-61 3"38

1.87 2.09 4"39

2.52 2-63 5"63

Method o f Goodrich and Cantow Mn Mw Mz Mz + 1

MOLECULAR WEIGHT AVERAGES CALCULATED WITH DIFFERENT THEORETICAL DISTRIBUTIONS

Method B Mz Mz + I

TABLE 4.

O ~v -t

.>

ta,

N

.>

OO

Two-parameter Functionsfor MolecularWeightof Polymers

813

fractionation data. In the polymer samples chosen for our comparison with experiment, the TPF which best describes the data is the exponential one. This is known to be the case for many but not all other polymer samples. It is impossible to predict beforehand which functional form of the possible TPF's will fit best a given polymer. The use of the linear combination avoids the need for a choice of one of them. The additional two parameters, introduced in the form of coefficients, serve to fit the experimental data in a more refined way than any of the TPF's. In the case of method A, the z and z + 1 averages of M are exactly given by the linear combination. This is a notable advantage over the TPF's because the width of the weight distribution WM is determined by the ratio /Q'z/37,~.~9~ In Fig. 1 we show that 3~/Mw has quite a different variation with )ffw//Q, for each of the TPF's available. The linear combination is able to adjust any pair of values of M z / M w and ,¢lw/~r, for a given polymer. As noted by Rehage and Wefers, (1°~ the simultaneous use of these two ratios is necessary to decide about the spread of molecular weight for a narrow-distributed polymer. In addition to this adjustment of molecular weights, method A also gives a better fit of the cumulative curve than any of the TPF's, as is shown by its lower mean quadratic deviation A. With method B, the purpose is to lower this mean quadratic deviation to its minimum possible value. This minimum A is well below the deviations of the single TPF's, showing that the linear combination improves appreciably the fitting of fractionation data. The search for the minimum A has to be done at the expense of relaxing the condition that the theoretical distribution should reproduce the z and z + 1 averages of M, which is the cornerstone of method A. However, the minimum A is accompanied by a closer description of the real 3,7~ and 3~r~+1 than that given by the individual TPF's (with the exception of M=+I r for sample R2). In fact, the linear combination obtained when method B is applied to these polymers is superior in its description of the first four averages of molecular weight, not only to the TPF's but also to the more sophisticated method of Goodrich and Cantow. t4'1~) The averages calculated using Goodrich and Cantow's method and reported by L6pez Madruga et al. ~6~ for their polymers are shown in Table 4 for this comparison. In the case of polymer R3, the least-squares fit (method B) produces a poor estimate of h7=+1. This is probably due to the fact that, in the fractionation of this polymer, the region of high molecular weights (M> ~ , ) contains fewer experimental points than the region of low molecular weights (M< ASt,). The tail of the distribution is thus ill-determined and gives unreliable values of the higher averages. We finally mention that the use of linear combinations of TPF's has been previously exploited by Koningsveld and Staverman/TM but with a purpose notably different from the one here considered. Instead of combining TPF's of different functional form and the same ~ , and r, Koningsveld and Staverman combined two TPF's of the same functional form and different ~ , and r; the distributions thus obtained are bimodal and are intended to represent the possible existence of two peaks in the differential curve. REFERENCES (1) L. H. Peebles, Molecular Weight Distributions in Polymers. Interscience, New York (1971). (2) M. Kubin, Colin Czech. chem. Commun. 32, 1505(1967). (3) A. M. Kotliar,J. Polym. Sci. A2, 4302 (1964).

814

E. SAIZ and A. H O R T A

(4) F. C. Goodrich, in Polymer Fractionation (Edited by M. J. R. Cantow). Academic Press, New York (1967). (5) L. H. Peebles, In Polymer Handbook, p. 11-429 (Edited by J. Brandup and E. H. Immergut). Interscience, New York (1966). (6) As defined in Handbook of Mathematical Functions (Edited by M. Abramowitz and I. A. Stegun). National Bureau of Standards, Washington D.C. (1964). (7) B. E. L6pez Madruga, J. M. Barrales-Rienda, and G. M. Guzmfin, Anal Quim. 67, 153 (1971). (8) G. V. Schulz, Z. phys. Chem. 1347, 155 (1940). (9) R. Koningsveld, Adv. Polym. Sci. 7, 1 (1970). (10) G. Rehage and W. Wefers, J. Polym. ScL A2, 6, 1683 (1968). (11) F. C. Goodrich and J. R. M. Cantow, J. Polym. Sci. C8, 269 (1965). (12) R. Koningsveld and A. J. Staverman, J. Polym. Sci. A2, 6, 305 (1968). R6sum6--On propose une combinaison lin~aire de fonctions exponentielle, logarithmique et normale d~crivant empiriquement les distributions de poids mol~ulaire de polym6res fractionn6s. Cette combinaison lin6aire contient quatre param~tres ajustables. Deux d'entre eux sontfix6s par les valeurs exp6rimentales de .~/, et ~ r . Les param~tres restant sont d6termin6s par deux m6thodes diff6rentes. L'une utilise ~/'~ et ~r,+l tandis que l'autre consid~re l'ajustement des moindres carr~s pour les donn~es de fractionnement. On donne un exemple num6rique de l'int6r6t de cette combinaison lin6aire en appliquant les deux m6thodes h un fractionnement de poly(m6thacrylate de m&hyle). Sommario--Si propone una combinazione lineare di funzioni esponenziali, iogaritmiche e normali per la descrizione empirica della distribuzione di pesi molecolari in polimeri frazionati. Questa combinazione lineare contiene quattro parametri variabili, di cui due sono determinati dai valori sperimentali di ~/, e -~/'we gli altri due con due altri differenti metodi. Un metodo fa impiego di 2Wz e -~lz+ 1, mentre l'altro 6 quello dei quadrati minimi dei dati di frazionamento. Si fornisce un esempio numerico dell'utilith di questa combinazione lineare applicando entrambi i metodi al frazionamento del poli(metil acrilato). Zusammenfassung--Es wird eine lineare Kombination der exponentiellen, logarithmischen und normalen Funktion ftir die empirische Beschreibung der Molekulargewichtsverteilung fraktionierter Polymerer vorgeschlagen. Diese lineare Kombination enthalt vier anzupassende Parameter. Zwei yon ihnen sind durch die expedmentellen Werte von .~/', und -~/'wfestgelegt. Die fibrigen Parameter werden nach zwei verschiedenen Methoden bestimmt. Eine benutzt Ji;/, und .~rz+ 1, wahrend sich die andere nach einer Anpassung an die Fraktionierungsdaten tiber die kleinsten Fehlerquadrate richtet. Ein numerisches Beispiel ftir die Brauchbarkeit dieser linearen Kombination wird gebracht durch die Anwendung beider Methoden auf die Fraktionierung von Polymethacrylat.