polymolecularity and polydispersity in molecular …

POLYMOLECULARITY AND POLYDISPERSITY INMOLECULAR WEIGHT DETERMINATIONS

HAN.S -GEORG ELIAS

Midland Macromolecular Institute, Midland, Mich. 48640, USA

ABSTRACTThe influence of polymolecularity on molecular weight determinations isreviewed. Several molecular weight methods give so-called mixed averages,i.e. averages which are based on more than one moment of the molecularweight distribution or on powers of the moment. Molecular weight averagescan be subdivided into simple, exponential resolved, order resolved and powerresolved averages. Mixed averages depend on the shape of the solute and itsinteraction with the solvent. Simple averages of molecular weight can becalculated from combinations of averages of different properties with the helpof the H-theorem. A related problem is that of the so-called polymolecularitycorrections. New experiments show a definite influence of the type and thewidth of the molecular weight distribution on the second osmotic virialcoefficients which is not predicted by present theories. Methods are given todistinguish between open, closed, molecule-based and segment-based associa-tions. The influence of the polymolecularity of unimers on the polydispersityof the resulting multimers is discussed for several types of association. Ingeneral, a sharpening effect is observed. Equations are given which may helpdetect polydispersity in open and closed association. Debye's treatment of

micellar solutions cannot be used to determine premicellar equilibria.

1. INTRODUCTIONAll synthetic and many biological polymers exhibit a more or less broad

molecular weight distribution (molar mass distribution)1, which influencesmany properties of macromolecular materials. Although the effects of thisdistribution have to be eliminated first before conclusions can be drawnon the effect of other quantities (such as composition, configurationalstatistics, conformational statistics, supermolecular structures, etc.) on cer-tain properties, astonishingly little work has been done to quantitativelyevaluate the effect of the molecular weight distribution on the variousproperties2. This is even true for the methods which are most directlyaffected by the molecular weight distribution, namely molecular weightdeterminations.

Molecular weight determinations are normally carried out by measuringcertain molecular weight dependent properties at various concentrations.Typical molecular weight dependent properties are, e.g., the osmotic pressure,the Rayleigh ratio, the sedimentation coefficient, etc. Because the numericalvalues of these properties are influenced by various interactions, the proper-ties have to be extrapolated to zero concentration. The extrapolation

115

HANS-GEORG ELIAS

procedures require theoretical guidelines, i.e. they require a molecular,therm9dynamic, hydrodynamic, etc., model of the solution.

It has been long known from statistical thermodynamics that the concen-tration dependence of the chemical potential of the solvent (and consequentlythe osmotic pressure) of solutions of non-electrolytes can be expressed asa power series with positive integers of the solute concentration3. Becauseof the relationship between chemical potential and molecular weight, asimilar power series describes the concentration dependence of apparentmolecular weights. This power series can be generalized to

(Mg)p; = M' + X(A2)avC + IJ(A3)avC2 + ... (1)

Mg is the q-average of the molecular weight, e.g. the number average (g = n),the weight average (g = w), etc. The apparent g-average molecular weight(Mg)app is the molecular weight calculated from experimental data at finiteconcentrations c with the help of an equation valid for zero concentrationonly. The second, third... virial coefficients (A2)av, (A3)av... representdifferent averages if different methods are employed. For colligative methods,one gets4

= w.w3.(A)1 (2)

and for weight average methods (light scattering, equilibrium ultracentri-fugation)46

=( (3)

where w, w, = mass fractions of species i andj, respectively. The coefficientsand f3 are also method dependent, giving = 1 for osmotic pressure

measurements, = 2 for light scattering experiments and =2f() forsedimentation equilibrium experiments7 where

f() = 1 + [M/12] + [M+1/M)2— 2(M+ 1M+2)2/M] (M)4/72O + ... (4)

The increase of the amount of molecules is directly given by the increase ofthe mass of solute in non-associating non-electrolyte solutions. This is nolonger true for associating solutes because less than a proportional number ofparticles are formed by an increase of the mass concentration. The relation-ship between mass and mole concentrations obviously depends on the typeof association. For all practical purposes, the extent of the association can beconsidered independent of the type and magnitude of the other interactions,e.g. solute/solvent interactions and/or intramolecular associations. In manycases the interactions are such that the association can be assumed to happenin a state similar to a theta state. We can thus write

(Mg); = (Mg)pg + ( cc(A2)/c2)+ ... (5)

where (Mg)app, is the concentration dependent association term.116

POLYMOLECULARITY AND POLYDISPERSITY

This paper will discuss the influence of polymolecularity on the type andmagnitude of the molecular weight averages Mg (see equation 1), on themagnitudes of the different second virial coefficients (see equations 2 and 3),and on the association of molecules (see equation 5), especially on thepolydispersity of the resulting associates (aggregates, micelles, etc.). Theeffects of polymolecularity on associated solutes are treated for thetasystems only, because the problem of the virial term in associating systemsis insurmountable at the present time. The approach will be mainly mathe-matical and based on certain physically most likely models, because thiswill allow a unified treatment. Little attempt is made to compare the mathe-matical predictions with experimental findings, mainly because the few,if any, data reported in literature are mostly incomplete for our purposes.

The problem of molecular weight and other averages has been recentlyreviewed2. Association processes have been discussed in general8' ,and withspecial emphasis on '° synthetic polymers12, sedimentationequilibrium13 and light scattering9. None of the review articles treats theeffects of polymolecularity and polydispersity adequately, if at all.

2. GENERAL ASPECTS OF POLYMOLECULAR1TY

Types of averagesThe distribution function of any property E can be described by a sufficient

number of moments. Moments t of the g-distributions of the propertiesE with respect to the origin are defined via

4(E) GE7 (6)

where q = order of the moment and

G1 g. = gg (7)

is the normalized statistical weight of the species i. The physical unit of amoment of a property E thus depends not only on the physical unit of theproperty but also on the order of the moment. The order may take any value,i.e. it may be a rational or irrational number.

An average, on the other hand, is defined in such a manner that the physicalunit of the average is always identical with the physical unit of the property.This requirement is of course met by a number of expressions. Hence, thereare a number of classes of property averages, e.g. simple averages, exponentialresolved averages, order resolved averages, etc. In each class, number,weight, length, area, etc., averages may be defined. The following discussionis restricted to molecular weight averages and moments.

Simple averages represent the first moment of the molecular weightdistribution curve:

Mg = G.M1 4')(M) (8)

They are always arithmetic averages. Most familiar are the number(G1 = x1 = mole fraction), weight (G1 = w1

= mass fraction) and z-averages(G1 = Z. = w1M1).

117

HANS-GEORG ELIAS

Exponential resolved averages are also based on a single moment. Becausethe order of this moment is different from one, the whole moment has to betaken to the inverse power of the order of the moment to yield an average

1/qMay =

( G.M7)= q))1/q (9)

Exponential resolved averages are normally named after the method, i.e.one speaks of viscosity average molecular weight, sedimentation averages,diffusion averages, etc. This nomenclature is, however, not sufficient becausethe average given by a certain method may also depend on the method ofevaluation and thus on the statistical weight used by the evaluation procedure.The well-known so-called viscosity average molecular weight (with q =a =exponent in the intrinsic viscosity/molecular weight relationship,[77] = KqM") is thus more properly called a mass (or weight) averageexponential resolved average because intrinsic viscosities of mixtures ofpolymers without specific interactions are always mass averages'6.

Order resolved averages are composed of two moments of the samestatistical weights which differ in their orders by one:

May = G1M + G.M = ,(q+ 1)/(q) (10)

Order resolved averages are quite frequent if molecular weights are deter-mined by the Svedberg method (combination of sedimentation and diffusioncoefficients) (see also below). The molecular weight from gel permeationchromatography is also an order resolved average (G = w and q = a )17

Geometric averages are normally classified as the square root of the proauctof two simple molecular weights with different statistical weights g and h:

May (MgMh) (11)

An example is the so-called M_1 average18, which has been shown to be ageometric average (see reference 2):

(mji>m1M[ 2) = (MM_J (12)

where m. = mass of species i. A closer inspection shows that the geometricaverages are a sub-class of the exponential resolved averages. Equation(12) can be written as

M1 = (WM2)-+ (13)

It is thus the exponential resolved weight average of the minus secondmoment of the molecular weight. Another example is the 'root mean square'average19, which is nothing but the exponential resolved weight averageof the second moment of the molecular weight2:

/Mrms = (MM) =

(Y wM) (14)

118


Power resolved averages are products of moments defined2 in such amanner that the exponents of the moments differ by one:

p p—iMay =

( G1M) (GMi)= (Mg)"(Mg_)' ° (15)

Many other averages can be defined on this basis. They all can be coveredby a general expression2°

1/p (16)May =(>G1Mr(1_h/G1M_1)

There are, however, a variety of averages which cannot be expressed bysimple product sums. Sedimentation equilibrium measurements in a densitygradient yield, e.g., molecular weight averages which contain exponentialexpressions21' 22:

GM exp [(—Mj/(2M)]M =

GM exp [(—M/(2M)](17)

where Ma is obtained from the half-width of the density distribution.

Statistical weightsThe most commonly used statistical weights can be subdivided into two

classes: entity-based and shape-based statistical weights. The class of entity-based statistical weights comprises the number statistical weight, the massstatistical weight, their higher and lower homologues (see below), and theirequivalents. The class of shape-based statistical weight consists of the length,area and volume statistical weights.

The class of entity-based statistical weights can be considered as derivativesof the numbers N1 of the species i or, of course, their amounts n or theirmole fractions x1:

n1 = Nj/NA (18)

x. = '>2 n1 = N1/ N1 = N1/N (19)

where NA is Avogadro's number and N is the total number of molecules.The next higher homologue is the mass

m1 = n1M1 = NIMI/NA (20)

or its equivalents such as the mass fraction w1 or the so-called weight con-centration (volume related mass):

w. = m11/rn = c1/c1 = cdc (21)

= m1/V01110 (22)

All other entity related statistical weights can be introduced by analogywith equation (20), i.e. as products of the statistical weight of next lower rankwith molecular weight. Examples are

'119

HANS-GEORG ELIAS

(n — 1)-statistical wt: (n — l) = n1M11 = mM2 = (m — 2) (23)

number statistical wt: = nM° = m1MI' = (m — l) (24)

mass ('weight') statistical wt: m1 = n.M = mM° = m1 (25)

z-statistical wt: = nLM = mM1 = (m + l) (26)

(z + l) statistical wt: (z + l) = nM = mM = (m + 2) (27)

The first expressions give the commonly used symbols for the differentstatistical weights; note that these are symbols and not mathematicaloperations. The second expressions relate amounts and molecular weights,the third expressions mass and molecular weight. The fourth expressionsgive alternative symbols. One may see that the mass can be taken instead ofthe amount as the base unit for entity-based statistical weights, or even thez-unit, etc.

If the species i are not molecularly homogeneous but exhibit a molecularweight distribution themselves, the molecular weights to be used in equations(23)—(27) should always correspond to the multiplying statistical weight23.An example is

(z + l) = z(M) = m(M)(M1) = etc. (28)

It must be mentioned that this statement can be generalized for any averageof any property24.

Shape-based statistical weights are lengths, areas and volumes or thecorresponding fractions. They are also related to moles and masses in thecase of simple bodies. The interrelationship between entity- and shape-based statistical weights depends, however, on the shape of the species. Forspheres, rods (with length L and constant and negligible radius R) and discs(with constant and negligible height H) with the surface A one gets, e.g.for the different masses if the densities p are constant:

spheres: rn = NV1p = N(4icR3)p = N.(p/3)RA1 (29)

rods: m1 = N1l'p = N1(mR2Ljp N1(p/2)RA (30)

discs: rn1 NJ'p = N.(irRH)p N(p/2)HA (31)

Polymolecularity parametersThe ratio of any two different molecular weight averages or any two

different moments of the molecular weight distribution can be used tocharacterize the width of a molecular weight distribution. The two quantitieshave to be chosen in such a way that their ratio is a sensitive and absolutemeasure for the width of the distribution.

In many cases, the ratio of two simple molecular weight averages is asensitive enough quantity for the characterization of the polymolecularity,e.g. the polymolecularity index,

= MW/MA (32)

or its equivalent, the inhomogeneity parameter25,

Un,,,, = (M/M) — 1 (33)

120


It can be easily shown that most other polymolecularity parameters such asthe ratio of two viscosity averages26 or the ratio of two different molecularweight averages from combined sedimentation and diffusion27 are, ingeneral, less sensitive than the polymolecularity index or the inhomogeneityparameter (see reference 2).

The standard deviation of the molecular weight distribution can be usedas a measure of the polymolecularity2835. As one can see from, e.g., thenumber standard deviation

cr, = (MWM — MJ = (U, (34)

it is more sensitive than the polymolecularity index. The standard deviationsare absolute measures for the width of Gaussian distributions, since thequantity (M ± a) always denotes that 68.26 per cent of the molecularweights are in this range and the quantity (M ± 2cr) always denotes 95.44per cent of the molecular weights, etc., independent of the position and thewidth of the Gaussian distribution curve. The standard deviation is, however,not an absolute measure for the width of other types of distribution, becausethe fraction of the molecular weight as characterized by a, and its multiplesis always a function of the width of the distribution36. No attempts havebeen made so far to introduce absolute measures for the width of distributioncurves for other than Gaussian distributions.

3. MIXED AVERAGES

Types of molecular weightA mixed average will be defined as an average which is not a simple

average as defined by equation (9). All mixed averages are thus characterizedby the existence of at least one moment with the order q 1. Typicalexamples are the molecular weight determinations from intrinsic viscosity/molecular weight relationships:

[ii] = KM" (35)

or from sedimentation and diffusion coefficients via the Svedberg equation:

f RTMay =[(1 —

i32p1)D(36)

Other equations leading to mixed averages May of molecular weight aresummarized in Table 1. Most property/molecular weight relationships canbe expressed by exponential equations similar to equation (35) for the wholemolecular weight range of interest:

E = KEMaE (E = [n], s, D, etc.) (37)

A theoretical justification is lacking in most cases, however. Exponentialrelationships can be even used ii these equations contain additive terms6063,or if the exponents aE are molecular weight dependent themselves64' 65

Analysis of the physical units (i.e. 'dimensional analysis') shows that theseexponents are related to each other by the so-called exponent rule50'6 6668

afl23aS(l+3aD) (38)

121

Tab

le 1

. Ph

ysic

al

equa

tions

and

gen

eral

exp

ress

ions

for

ave

rage

s fo

r m

olec

ular

wei

ght

equa

tions

giv

ing

mix

ed

aver

ages

. ['i

] =

intr

insi

c vi

scos

ity (a

lway

s as w

eigh

t av

erag

e); <

s,,>

, <Sm

,,) =

wei

ght a

vera

ge se

dim

enta

tion c

oeff

i-

cien

t and

aver

age f

rom

the m

axim

um of

the

diff

eren

tial s

edim

enta

tion c

oeff

icie

nt d

istr

ibut

ion c

urve

; (D

v>, <

Da>

=

wei

ght a

nd ar

ea a

vera

ge d

iffu

sion

coe

ffic

ient

; <

MG

> =

mol

. wt f

rom

the

max

imum

of a

gel

per

mea

tion c

hrom

ato-

gr

aphy

dia

gram

; K

,, K

,, K

, KD

, K

, = ca

libra

tion

cons

tant

s in

prop

erty

/mol

ecul

ar w

eigh

t re

latio

nshi

ps si

mila

r to

equa

tion

(38)

; a,, a

,, a,

aD, a

, = e

xpon

ents

in th

ese r

elat

ions

hips

; R =

gene

ral

gas

cons

tant

, T =

ther

mod

ynam

ic

tem

pera

ture

, 2

= p

artia

l sp

ecifi

c vo

lum

e of

the

sol

ute;

p1

= d

ensi

ty o

f th

e so

lven

t; N

A =

Avo

gadr

o nu

mbe

r;

k =

Boltzmann c

onst

ant;

fi =

constant;

= vis

cosity of

sol

vent

; A,

B =

calibration c

onst

ants

; l'

= el

ution

volu

me;

w =

mass f

ract

ion

Ref

eren

ces fo

r ge

nera

l Sy

mbo

l Ph

ysic

al eq

uatio

n G

ener

al ex

pres

sion

for a

vera

ge

phys

. eq.

ex

pres

sion

M,

=

=

M,,,

= (

Smax/K)lIa

MD

=

MDA

= (DA/K,)1D

z C,..

,

C)

C)

tTI

/ 1/

a,I

WIM

3/(2

—a,

)

( W.M

23)

— 3/

(1 +aD)

w.M

+aD

)/3)

6/(1

+a,)

( wM

+an

)/6)

37—

40

41

—

42

MS

D =

RT

(1 —

MS

DA

= R

T(1

— i3

2p1)

sD;'

=

—

MD

= (

kTf3

rj 1)

3D; 3[

77] -1

MD

A =

(kT

flj 1)

3D; 3

[i]1

M0

= lo

g M

0 =

A —

Bl'

( w.M

2 a,

,) /3

) ( w.M

1 +

an)/

3)

1

( W

1M2a

73) ( W

' +af

l)/6)

( w1M

2 _a

n)/3

)( w

.M7n

)

( w1M

2'7)

( w

M1"

+a)

I3)

( w1M

ai) - '( w

M +

a7)/

6)

( wM

+a)

( WM

)'

43

44-4

6

43

47—

49

47,

49

47, 4

9 51

17

I%J

44, 5

0

44, 5

0

50

Tab

le 2

. M

ixed

ave

rage

s of

mol

ecul

ar w

eigh

t fo

r po

lym

ers

with

Sch

ulz—

Flor

y and

loga

rith

mic

nor

mal

dis

trib

utio

ns; 2

= co

uplin

g pa

ram

eter

of t

he S

chul

z—

Flor

y di

stri

butio

n as

def

ined

by

<M

>/<

M>

= (2

+ 1

)/2

Ref

eren

ces

for

Mol

ecul

ar w

eigh

t av

erag

e fo

r Sc

hulz

—Fl

ory

log

norm

al

log n

orm

al d

istr

. di

str.

di

str.

M(M

/Mj'

+a)

/2

52—

54

55,

56

) M

exp

[(an

— O

.5)(

M/M

J]

57

57, 5

8

Sym

bol

Schu

lz—

Flor

y di

str.

M /T

(2 +

1 +

M

JT

(2+

1) I

M 13

2 +

1 +

aq

M

3

M I

F'(2

+ 1

) \3

/(1

+fl

) M

D

= 2

T{(

32 +

2— a

)/3}

)

M /r

{.(6

2 +

7 +

a)/

6}

MD

A

= 2

f(2

+ 1

) )6

/(1 +

a)

M /3

2 +

2 —

a

MSD

2

3

M IJ

T{(

32 +

5 +

a)/3

}f2{

(62

+ 7

+ a)

/6})

M

SDA

=

T

3(2

+ 1

)

M If

*{(3

2 +

5 —

a)/3

}T(2

+ 1

+ an

))

F2(2

+ 1

)

M (

32_+

1 +

a)(2

_Jn2

Ir(2

+ 1

+ a,

,'\+

M

a2

\ 32

I'(

2+1)

)

(___

['(

A+

1)

MD

—

2

\T{(

32 +

2 —

a9)/

3}F

(2 +

I +

an)

)

Mn(

F6{(

62 +

7 +

a)/6

} \

M D

A?

= 2

T5(

2 +

I)(F

+ 1

+ a

n))

M

M6

=—

,-(2

+1-

f-a)

59

59

46

M(M

IM) —

2a,,

+ 9)

/12

50

2a7±

5)/

6 50

2

5ac—

2)/

36

57

57,

58

50

2

MjM

/M)5

— 2a

— 7

a)/1

2 50

2

ni

C

Ci

ni

17


Equations (37) and (38) allow an analysis of the types of molecular weightaverages of physical equations typically leading to mixed averages. Examplesare equations (35) and (36); others can be found in Table 1. One simplyreplaces the molecular weight dependent physical quantities like the intrinsicviscosity, the sedimentation coefficient, etc., by their averages (e.g. weightaverage), expresses the properties of the species i by the correspondingexpressions for their molecular weight dependence (see, e.g., equation 37),and finally replaces the individual exponents aD, a, etc., by the exponenta with the help of equation (38). The resulting expressions are then convertedinto averages or moments. Some of these expressions are summarized inTable 1. They can be replaced by specific functions if the type of the molecularweight distribution is known (Table 2). The resulting molecular weight

Figure 1. Influence of the parameter a as a measure for the solute shape and the solute/solventinteraction on the numerical values o several mixed average molecular weights for a polymer

with a Schulz—Flory molecular weight distribution and a MW/Mfl 2(i.e. = 1)

125

a.11

HANS-GEORG ELIAS

average thus depends on the average of the properties (e.g. intrinsic viscositysedimentation coefficient, etc.), the type and the width of the molecularweight distribution, and the shape of the solute and its interaction with thesolvent. The influence of the latter, e.g. as given by the exponent a of theintrinsic viscosity/molecular weight relationship, can be especially pro-nounced (Figure 1). Sometimes the mixed averages are identical with simple

Table 3. Averages and moments of molecular weights (molar masses) from Svedberg and Mandel-kern—Flory—Scheraga equations for measurements of random coils in the unperturbed state(a = ).D = i) 2 is the so-called area average of the diffusion coefficient. The numericalvalues refer to a Schulz—Flory 'normal' distribution with MW/Mn = 2and M, = 100000 g mo11

Mixedaverages Molecular weight averages or moments

Numerical value(g mol 1)

MSDMSD

MMj/j4100000150000

MSDMSDAM,,MDMDAMM0

MpW(JLW)2(ii)2 = M

((*))6/(+)())2 = M()(/2)

200000170646176 716108 074159126176716250000

averages. A few examples are shown in Table 3 for the exponent a, e.g.for random coils in a theta solvent. Very different molecular weight averagesmay thus result if the property average is changed (Table 3). Two facts areof special interest. Although many physical equations of those shown inTable 1 clearly give 'absolute molecular weights' (i.e. the molecular weightcan be calculated from experimental data without calibration), the resultingmixed averages may depend on the solvent. One may note furthermore thatthe combination of the weight average of the sedimentation coefficient and theintrinsic viscosity gives directly the viscosity average of the molecular weightfor random coils in the unperturbed state69. This molecular weight Msflcan thus be used directly to establish intrinsic viscosity/molecular weightrelationships without the usual polymolecularity corrections.

Very few experimental data have been reported in literature which canbe used to check the predictions made by this simple approach. Molecularweight determinations on a poly(styrene) sample in cyclohexane at 34°C(theta conditions) yielded M = 423000, M = 239000 and M = 60000g mol 1, i.e. the distribution can be represented by a Schulz—Flory distribu-tion with 2 = -. From these data, one can calculate M, = 161000 (seeTable 2), whereas experimentally a value of 167000 g mo'1' was found.Kotera, Saito and Takemura7° determined many mixed averages of molecu-lar weight for a mixture of two samples and compared them with the predictednumerical values. The agreement was good to fair in most cases. It must bementioned that the unknown molecular weight distribution of the two parentsamples may have affected the results to some extent, since the authorsassumed the parent samples to be molecular homogeneous.

126


Simple averages from combinations of propertiesThe previous section was concerned with the problem of which molecular

averages result if simple property averages are inserted in one of the physicalequations given in Table 1. In many cases, however, one is interested in theopposite problem: how to get a simple molecular weight average from acombination of property averages.

Early papers were mainly concerned with measurements in the thetastate (Table 4), although a general solution for the problem of determinationof weight average molecular weights from sedimentation coefficient/mole-cular weight relationships has been known for some time34' 71

Table 4. Simple molecular averages from combinations of property averages in complexmolecular weight equations for measurements in the theta state. The symbols have the same

meaning as in Table 1

Molecular weightaverage Proper combination Ref.

M K' (ws2) 71,72

M K1(ws) 73

M

M

ff RT(1 — i32p, ) (

[Na?119(1—

132p1) 1]( ( wist) ['i]

74

75, 76

A general solution for the problem of finding simple averages of molecularweight from combinations of arbitrary averages of other properties has beenpresented recently77 and utilized for the calculation of the weight averagemolecular weight from sedimentation and diffusion coefficients78.

Polymolecularity correctionsExact correlations between two different properties can be established

only if the proper averages of these properties are used. A well- and long-known example is the correlation between intrinsic viscosity and molecularweight where the intrinsic viscosity is a simple weight average'4'6 and themolecular weight is the so-called viscosity average'6, which is an exponentialresolved weight average2.

Polymolecularity correction factors have to be used if other than theproper property and molecular weight averages are used to establish cor-relations. These correction factors can range up to several hundred per centeven for distributions that are not very broad, and depend on the type of theproperty/molecular weight relationship, the type of the molecular weightdistribution, the shape of the polymer and the polymer/solvent interaction.

127

HANS-GEORG ELIAS

Polymolecularity corrections are most dramatic for the so-called Baumannequation79:

(<R2>/M) = + BM (39)

KM* + BM2 (40)

where <R2> = mean square radius of gyration; K0, B = system specificconstants. Normally, the radii of gyration are determined as z-averages andthe molecular weights as weight averages. Polymolecularity correctionfactors can then be calculated for either the constants K0 and B8° or (better)for the two variables <R> and MW81. Taking the z-average of equation (40)

= Z1(R = K ZM + B ZM

and comparing this expression with equation (39) written for weight averagemolecular weight and z-average mean square radius of gyration gives thepolymolecularity correction factors81

= K + B(Mq) (42)

with

=( ZR3) (ZiM)2/<Rz>(ZM?)

(43)

qM = ( ZM) /MW( zM) (44)

0E5EU• 4

,%'

1

0 500 1000O 1/2

<M>w)w

Figure 2. Influence of polymolecularity corrections on the molecular weight dependence ofreduced radii of gyration of poly(a-methylstyrene)s in toluene at 25°C (from ref. 83). Uncorrecteddata (q = 1; lM = 1) are shown as open symbols, i.e. Schulz—Flory distributions (cJ), logarith-mic normal distributions (Lx) and 'narrow' distributions (0), whereas data corrected for poly-molecularity are given by full symbols, i.e. Schulz—Flory distributions (N), logarithmic normaldistributions (A) and 'narrow' distributions (•), corrected under assumption of a Schulz—

Flory distribution

128


Equations (43) and (44) contain moments which are not easy to determineexperimentally. They can be replaced by experimental quantities if the typeof the molecular weight distribution is known.

Figure 2 compares experimental results on poly(-methylstyrene)s intoluene at 25°C with and without polymolecularity corrections. The dramaticinfluence of the polymolecularity corrections on the properties of sampleswith a logarithmic normal distribution of molecular weights is clearly seen.Many papers in literature contain remarks on polymolecularity correctionfactors; most of them are improper. Correct polymolecularity correctionfactors were given for the Kuhn—Mark—Houwink—Sakurada equation(equation 35)8O 82, for the Burchard—Stockmayer—Fixman equation80,for the Cowie relation between friction coefficient and molecular weight8°and for the above-mentioned Baumann equation80'81' 83• Polymolecularitycorrection factors were reviewed2 and tabu1ated84 in detail.

4. VIRIAL COEFFICIENT AVERAGES

The definitions of the second virial coefficient averages, equations (2) and(3), predict an (A 2)OP (A2)Ls which has been verified experimentally8 5—88In the case of mixtures of two molecular homogeneous polymers, equations(2) and (3) can be reduced to

(4\OP_j'4\ 2 1(A\ (A\ 2 4— 2flW + flJWWj + 2)JJWJ

(A2)" =[(A2)1Mw + + (A2)Mwfl/M (46)where (A2)1 and (A2) are the two second virial coefficients for the molecularhomogeneous polymers i and j, w and w the corresponding mass fractionsand (A2) the so-called cross virial coefficient.

Equations (45) and (46) predict maxima in the functions = f(w3) andAjS = f(w .), if(A2)11 > (A2) or (A2)> (A2) (see reference 89). The secondvirial coeMcients decrease with increasing molecular weight in a homo-logous polymer series according to

A2 = KAM_A (47)

The predicted maxima have been found by some authors9092, but not byothers93' Several reasons might be responsible for this discrepancy:(a) the polymers i and j were not homogeneous and the averaging mightthus have been improper; (b) the evaluation of the second virial coefficientswas incorrect, e.g. by using a certain relationship between the second and thethird virial coefficient; and (c) the maximum may lie at very low values ofw and thus be difficult to detect.

The effect of the type and width of the molecular weight distribution wasinvestigated only recently95. Many samples of poly(c-methylstyrene) weremixed together to obtain a preselected distribution function, and with theweight average molecular weight held constant (M = 80000 g mol ').The second virial coefficients from light scattering measurements wereabout constant and independent of the varying number average molecularweight and the type of the distribution. With increasing polymolecularityparameters, the osmotic second virial coefficients of the mixtures with

129

PAC—43-1/2-F

HANS-GEORG ELIAS

Schulz—Flory distributions decreased, whereas the corresponding osmoticsecond virial coefficients of mixtures with logarithmic normal distributionsincreased much more rapidly than predicted by the hard sphere approxima-tion theory (Figure 3). Replacement of the hard sphere approximations by asoft sphere model96, i.e. a model with overlapping random coils, does notlessen the disagreement between experimental and theoretical second virialcoefficients. The observed discrepancy might well be due to a breakdownof equation (47) for low molecular weights. Further theoretical and experi-mental work is clearly needed in this area.

0E

E0QCN

0

Figure 3. Dependence of osmotic second virial coefficient on the polymolecularity indexM/M for poly(x-methylstyrene)s in toluene at 25°C with constant weight average molecularweight M 80000gmol . The symbols indicate samples with logarithmic normal distribu-tion of molecular weights (0), Schulz—Flory distributions (•), a narrow distributed sample withunknown type of distribution function () and the data point from the calibration curve for ahypothetical molecular homogeneous sample (0). The dotted line ( ) indicates the predic-

tion of the hard sphere approximation. (From ref. 95)

5. INFLUENCE OF POLYMOLECULARITY ON ASSOCIATION

Distinction between types of associationAssociation equilibria can be conveniently described by two limiting

cases and combinations thereof: open and closed association89. Openassociation is defined as an association of unimers with molecular weightM1 to an unlimited number of multimers:

2M1 ± M11

M11 + M1 ± M111 (48)

M111 + M1 M1, etc.

130

LN

LN.SF

3 3.5


Closed association, on the other hand, is characterized by an equilibriumbetween two distinct species only, unimer and r-mer:

rM1 ± Mr (49)

Both of these limiting cases can be distinguished easily by their charac-teristic concentration dependence of apparent molecular weights in the lowconcentration range (Figure 4). Only closed associations show a characteristic'critical micelle concentration' (cmc). Distinctions between closed and openassociations are less easily made if the whole concentration dependence ofthe apparent molecular weights is unknown, especially at low concentrations.In this case, both the concentration dependencies of the inverse apparentmolecular weight for open and closed associations look like hyperboliccurves (Figure 4).

2O i3i1 0.3 0.5 0.7

1 ____..J—cmc

0.8

aaci

A 0.1.

0.2

0 0.003 0.00/.

c,g cm3

Figure 4. Concentration dependence of the normalized inverse apparent g-average molecularweight in the theta state for molecule-based closed associations (r = 21; = iO' (dmamol 1)20; (M1) = (M1) = (M1) = 200 g mol 1) and the equivalent open association withK0 = (K)'°_ 1) Wr = Cr/C = mass fraction of the r-mer. Solid lines correspond to (hypo-thetically) experimentally accessible data, broken lines to data predicted by the model. The'critical micelle concentrations' cmc depend on the method and do not correspond to a 'first

appearance' of micelles. (Adapted from ref. 97)

Open and closed molecule-based associations (see below) can bedistinguished from each other, however, if, e.g., the apparent weight averagemolecular weight is plotted against the inverse number average molecularweight97 (see Figure 5). For closed associations, straight lines are observedfor theta systems, since

(Mw) app,0 <M1> + r<M1> — r<M1)(M),0 (50)

131

n.open

n,ctosed

0.001 0.002

HANS-GEORG ELIAS

Concave lines are observed for closed associations in non-theta systems.On the other hand, this plot always gives convex lines for open associationsin both theta and non-theta systems, as shown by the following equationfor open associations of rods under theta conditions (see reference 98):

(Mw) app,0 = — 2c(M1> + 2(Mn) app,0 (51)

Similar relationships can be derived for apparent z-averages. For closedmolecule-based associations, one obtains

(Mz)app,o <M1> + [1 + 2<M1> — — <M1>]-1

[1 — <M1>( w) app, ] ( )and for open molecule-based associations

(Mz)app, 0(Mw) app,0 = — ()<M1>] + () (53)

3

0E

20C

o 1

0 .1 2 3 5iü(M0)

Figure 5. Apparent weight average molecular weights in the theta state as a function of theinverse apparent number average molecular weight in the theta state for closed and openassociations with the same molecule-based open association equilibrium constant K0. Dataof Figure 4: 'K0 = ('K)11°_ 1); 'K = iO (dm3 mol 1)20; r = 21; (M1), = (M1),, = (M1), = 200

gmol . Broken lines correspond to theta state conditions, solid lines to end-to-end associationof rigid cylindrical rods with a radius of 5 x 8 cm and a specific volume of 1 cm3 g . The

concentration increases in the direction of the arrows. (From ref. 97)

Both open and closed associations can be either molecule- or segment-based.

A molecule-based association is defined as an association of moleculeswhose number of associogenic sites is independent of molecular weight. Acommon example is association via the two end-groups of-linear molecules.It is convenient and in many cases justified by either. experiment or chemicalintuition, to assume the equilibrium constants to be independent of theparticle size (principle of equal physical reactivities). This assumption

132

closed


allows the treatment of open associations with only a single equilibriumconstant.

A segment-based association is defined as an association of moleculeswhose number of associogenic sites increase linearly with the molecularweight. An example might be the association of partially stereoregularmolecules whose number of stereoregular segments increase with molecularweight.

Molecule- and segment-based open associations can be distinguished incertain cases, particularly if statistical weights of higher rank are used (seereference 98). The distinction is possible because the relation

(WK) = r(r — 1) 1<MI> 1(K0) = Q<M1> '(K0) (54)

varies with the average numerical values of r, which in turn varies withconcentration. The factor Q is equal to 2 for a unimer/dimer associationand approaches 1 for the limit of infinite r. wK0 and K0 are the equilibriumconstants of open association on a mass or mole basis, respectively, wherec = mass concentrations and [M] = mole concentrations:

WK Cr/(Cr_ 1c1) (54a)

K0 [Mr]/([Mr_ ] [M1]) (54b)

Molecule- and segment-based closed associations cannot be distinguishedfor a given sample since there is only one r, which, by definition, staysconstant over the whole concentration range. If, however, a molecularhomologous series is available (i.e. a series of compounds with the samephysical reactivity but different polymolecularities), then a distinction canbe made on the basis of their polymolecularity/polydispersity relationships(see below).

Influence of polymolecularity on polydispersityAn r-mer can be formed by any r unimer molecules of a polymolecular

unimeric compound. The mathematical treatment is especially simple if thephysical reactivity is independent of the molecular weight. The relationshipsbetween the g-averages of the molecular weights of unimers and multimersobviously depend on type of association, i.e. whether it is a molecule-basedor a segment-based association.

For molecule-based associations, the number average of the r-mer isjust r times the number average molecular weight of the unimer:

<Mr>n = r<M1> (55)

The weight average molecular weight of the r-mer is, however, not a simplemultiple of the weight average molecular weight of the unimer98

<Mr>w = <MI>W + (r — 1)<M1> (56)

and similarly for the z-average98

<Mr>z = <MI>Z + (r — 1)<M1>{1 + [2<M1> —

— <M1>] [<M1> + (r — 1)<M1>]1} (57)

133

HANS-GEORG ELIAS

For segment-based associations, general expressions have been derivedfor the weight and the z-average of particle weight of the r-mers98, whereasfor the number average a closed expression exists for the case of Schulz—Flory distributions only98:

<Mr>n = <M1> + (r — 1)<M1> (Schulz—Flory only) (58)

<Mr>w = r<M1> (59)

<Mr>z = <M1> + (r — 1)<M1> (60)

From equations (55H60), the following equations can be derived for therelations between the polymolecularity parameter <M1>/<M1> and thepolydispersity parameter <Mr>w/<Mr>n.For molecule-based associations:

<Mr>w/<Mr>n = 1 + r — 1] (61)

For segment-based associations with a Schulz—Flory distribution of unimers:

<Mr>i.J<Mr>w = 1 + r1[(M1>,.J<M1> — 1] (62)

or

<Mr>w/<Mr>n = [<M1>/<M1)]/[1— (1 — r)(<Mi>/c(Mi>)] (63)

Diagrams of these polydispersity/polymolecularity relationships (Figures 6and 7) show that association is accompanied by a pronounced sharpeningof the distributions. This effect has the same origin as the one observed forthe growing of many chains from one common growth centre99: the distribu-tions are transferred into one particle and the distributions over all particlesare thus more homogeneous. The effect is of course the more pronouncedthe higher the association number r. Segment-based associations always

1.5

1.4

1.3

1.2

1.1

2

(Mi)/ (M1),,

Figure 6. Influence of polymolecularity index (M1)/(M1) on the polydispersity index (Mr)w/(Mr)nfor molecule-based associations for various association numbers r. The relationships are

independent of the particular type of molecular weight distribution. (From ref. 12)

134

3 4 5

Figure 7. Influence of polymolecularity index (M1)/(M1), on the polydispersity index (Mr)wI(Mr)nfor segment-based associations for various association numbers r. The relationships are

independent of the particular type of molecular weight distribution. (From ref. 12)

Table 5. Influence of polymolecularity parameter <M1>/<M1>. and association number r on thepolydisperity parameter <Mr>wI<Mr>n

<M1>/<Mi>. r<Mr>wI<

Molecule-basedMr>nSegment-based

1.11.1

210

1.0501.010

1.0481.009

1.12

1002

1.0011.500

1.0011.333

22

10100

1.1001.010

1.0531.005

1010

210

5.5001.900

1.8181.099

10 100 1.090 1.009

6. POLYDISPERS1TY OF ASSOCIATED SOLUTES

Detection of polydispersity in open associationsOpen associations automatically lead to a polydispersity of the r-mers,

whether the unimer is molecularly homogeneous or polymolecular. It135


I;

r=4

5

10

2

20

3

(M1)/(M1 )

50

show more sharpening than molecule-based associations (Table 5). Thepolydispersity parameter increases with the polymolecularity parameterfor both molecule- and segment-based associations. In the latter case, butnot in the former, an asymptote of <Mr>w/<Mr>n = r/(r— 1) is reached foran infinitely high polymolecularity parameter.

HANS-GEORG ELIAS

remains to show that the polydispersity of the associate corresponds to theone predicted for an open molecule-based association with equal physicalreactivity of the unimers and r-mers.

In this particular case, one obtains for the concentration dependence ofthe different apparent molecular weight averages in the theta state98:

[(Mn) app, = <Mi>n + (nKo)<Mi>n[c(Mn);, ] (64)

[(Mw) app, ] = <M1> + 2(nKø)<Mi>n[C(Mn)p, ] (65)

[(Mz)app, 0] = (<M1> + 3(ZKø)<Mi>n[C(Mn), 0])<M1>[(M);, oil+ 3CK0)<M1>[c(M)p1p 0] (66)

Introducing the apparent association numbers (with g = n, w, z, etc.)

(Zg)app, 0 (Mg) app, oI<Mi>g (67)

one arrives at[(Zw)app, 0] = 1 + 2(<Mi>wI<Mi>n)[(Zn)app,o — 1] (68)

and

[(Zz)app, 0(Zw) app, ] = 1 + — 1] (69)

By plotting the quantities in brackets against each other, a straight lineshould be observed, if the data can be described by the model of an openmolecule-based association with equal physical reactivity of the species.Equation (68) resembles (but does not equal) the relation between weightand number average degrees of polymerization for condensation polymeriza-tions (see, e.g., reference 100), i.e. for the so-called most probable distributions(a special type of a Schulz—Flory distribution)

= 1 + 2(<X>n — 1) (70)

Comparison of equations (68) and (70) shows that the presence of a poly-molecularity of the solute, expressed by <MI>W/<MI>, increases the apparentpolydispersity of associating solutes relative to the polymolecularity ofpolycondensates. In the limiting case of molecular homogeneous unimers ormonomers, both equations become identical.

For segment-based associations, the concentration dependencies of theapparent weight and z-averages of molecular weight in the theta state can begiven with complete generality98:

(Mz)app,o = <M1> + 2C'K0) (M1)c (71)

(Mw) app,0 = <M1> + (wK0) (M1)c (72)

whereas the concentration dependence of the apparent number averagemolecular weights can be described by a simple equation only for the case ofa molecular homogeneous unimer:

(Mn) app,0 = <Mi>n(wK0)c[ln(1 + (wK0)c] —1 (73)

The combination of equations (71) and (72) leads to

(Zz)app,o = 1 + 2[(Zw) app,0 — 1] (74)

136


Detection of polydispersity in closed associations

The simplest model for a closed association assumes that only unimer andr-mers coexist in equilibrium. Because the relative concentration of thesetwo species varies with the total concentration, weight average methodswill invariably give higher apparent molecular weights than number averagemethods in the concentration range between the 'critical micelle concentra-tion' and the usually employed highest concentrations (see also Figure 4).A difference in different apparent molecular weights can thus a priori notbe interpreted as a polydispersity of the associates.

At very high concentrations, all apparent molecular weight averagesbecome identical for measurements in the theta state, if the simple unimer/r-mer equilibrium applies for. molecularly homogeneous unimers. Thesituation again becomes complicated if the second and higher virial termscannot be neglected.

For polymolecular unimers undergoing a simple unimer/r-mer associationin the theta state, the ratio (Mw)a p o/(Mn) app will equal the polydispersityparameter <Mr>w/<Mr>n at very ciigh concentrations. As may be seen fromTable 5, the polydispersity parameter is always smaller than the poly-molecularity parameter of the unimer and very often indistinguishablefrom 1 within the limits of error of the experiment. It is thus doubtful whetherany information on polydispersity or polymolecularity can be drawn fromthis type of experiment, particularly if the whole concentration dependenceof apparent molecular weights is unknown and/or virial coefficients have tobe considered.

Multiple equilibria in closed associationsThe previous discussion was concerned with simple closed associations, i.e.

associations between a unimer and a single r-mer. Combinations of differentclosed associations or of a closed and an open association may occur, of course.A very simple and often-suggested case is the so-called premicellar equili-brium. This name is given to a combination of two consecutive equilibria, e.g.

qM1 ± Mq; Kq = [Mq]/[Mj] (75)

r/qMq P Mr; Kriq = [M]/[Mq]rI (76)

q is generally considered to be small, e.g. 2 or 3, whereas r is high, typicallybetween 10 and 100. Any other definition can be used, too, e.g.

qM1 ± Mq; Kq = [Mq]/[M1] (77)

rM1 ± Mr Kr = [Mr]/[Mi]r (78)

All definitions lead to

[Mr]/[Mi]r = (Kriq) (1KqyI (79)

Equations (75), (76) or (77), (78) should in principle lead to two-step curvesin the (Mg)app, = f(c) diagrams. Whether these two-step curves can beobserved experimentally will of course depend on the relative magnitudeof the two association numbers q and r and the two equilibrium constants

137

Tab

le 6

. Ass

ocia

tion

num

bers

r an

d eq

uilib

rium

cons

tant

s K0

of si

ngle

-ste

p cl

osed

ass

ocia

tion

of m

acro

mol

ecul

es at

25°

C. A

U a

ssoc

iatio

n num

bers

and e

qui-

lib

rium

con

stan

ts ar

e av

erag

ed ov

er s

ever

al m

etho

ds, w

here

app

licab

le

<M

I>W

Po

lym

er

Solv

ent

g m

ol'

<M

I>

g m

o11

r K

0 dm

3 m

ol'

1og"

K

Ref

.

log[

(dm

3 m

ol 1)

1] c-

Hyd

roge

n-o-

hydr

oxy-

po

ly(o

xyet

hyle

ne)

Cd4

—

an

y 24

12

cx-H

ydro

gen-

o-hy

drox

y-

poly

(oxy

ethy

lene

) be

nzen

e —

an

y 11

12

x-H

ydro

gen-

o-hy

drox

y-

105—

2630

poly

(oxy

ethy

lene

) di

oxan

e —

an

y il

12

cL-H

ydro

gen-

w-h

ydro

xy-

poly

(oxy

ethy

lene

) ac

eton

itrile

—

an

y 0.

7 —

12

tT

I

poly

('y-b

enzy

l-L

-glu

tam

ate)

di

oxan

e 33

400

3150

0 25

.0

1330

00

123

101

poly

(y-b

enzy

l-L

-glu

tam

ate)

di

oxan

e/he

xane

33

400

31 5

00

18.7

12

1 50

0 90

10

2 C

po

ly(y

-ben

zyl-

L-g

luta

mat

e)

diox

ane/

hexa

ne

1540

00

1320

00

3.5

398

100

14

102

poly

(y-b

enzy

l-L

-glu

tam

ate)

di

oxan

e/he

xane

22

2000

20

2000

5.

4 81

1 10

0 26

10

2

poly

(y-b

enzy

l-L

-glu

tam

ate)

ch

ioro

form

/hex

ane

3340

0 31

500

12

.0

115 8

00

55.7

10

2

poly

(y-b

enzy

l-L

-glu

tam

ate)

ch

loro

form

/hex

ane

1540

00

1320

00

4.3

3514

00

18.3

10

2

poly

(y-b

enzy

l-L

-glu

tam

ate)

ch

loro

form

/hex

ane

2220

00

2020

00

5.5

1000

000

27

102

poly

(7-b

enzy

l-L

-glu

tam

ate)

ch

loro

form

/met

hano

l 33

400

3150

0 16

.0

7360

0 73

10

2

poly

('y-b

enzy

l-L

-glu

tam

ate)

1,

2-di

chlo

roet

hane

/met

hano

l 33

400

3150

0 12

.0

5340

0 52

10

2

poly

(y-b

enzy

l-L

-glu

tam

ate)

1,

2-di

chlo

roet

hane

/met

hano

l 33

400

31 5

00

9.4

6170

0 40

10

2

Tab

le 7

. Ass

ocia

tion

num

bers

r and

equ

ilibr

ium

cons

tant

s "K

0 of

sin

gle-

step

clo

sed

asso

ciat

ions

for s

ever

al d

eter

gent

s in

wat

er.

All

asso

ciat

ion

num

bers

and

equi

libri

um co

nsta

nts a

re av

erag

ed o

ver s

ever

al m

etho

ds, w

hen

appl

icab

le

Uni

mer

T

emp.

, St

ruct

ure

<M

I>W

r "K

0 dm

3 m

ol1

log

"Kr

Ref

. lo

g[(d

m3

mol

1)r

_ 1]

.u 0 t-

'-< <

.o

C8H

17—

C6H

1106

C

8H17

—C

6H11

06

C8H

17—

C6H

1106

C

8H17

—C

6H4-

--(O

CH

2CH

2)20

0H

C9H

19—

C6H

4——

(OC

H2C

H2)

420H

C

9H19

—C

6H4—

(OC

H2C

H2)

420H

C

9H19

—C

6H4—

(OC

H2C

H2)

420H

C

9H19

—C

6H4-

-(O

CH

2CH

2)62

0H

C9H

19—

C6H

4——

(OC

H2C

H2)

620H

1 10

0 21

55

2155

21

55

3855

38

55

312

312

312

—

1 81

3 18

13

1813

29

70

2970

20

30

50

25

20

30

50

20

30

68:3

84

.1

72.0

32

.4

18.6

20

.8

25.2

11

.0

13.7

31.2

42

.8

44.2

12

2 10

38

2401

29

55

794

1732

99.3

13

1.9

116.

4 65

.3

53.0

63

.6

84

29

45.6

103

103

103

104

105

105

105

105

105

C9H

19—

C6H

4—(O

CH

2CH

2)62

0H

CH

2=C

(CH

3)—

CO

N(R

)—(C

H2)

10—

CO

X

R=

CH

3 X

=(O

CH

2CH

2)15

0CH

3 R

C2H

3 X

=(O

CH

2CH

2)15

0CH

3 R

=C

3H7

X=

(OC

H2C

H2)

150C

H3

Ri-

C3H

7 X

=(O

CH

2CH

2)15

0CH

3 R

CH

3 X

ON

a R

C2H

5 X

=O

Na

RC

3H7

XO

Na

3855

29

70

937

1020

99

0 98

3

50

40

40

40

40

40

40

40

15.0

20

31

38

31 7 16

12

2683

336

430

474

501

147

54

152

48

48

79

99

81

13

26

24

105

106

106

106

106

106

106

106

u

Prot

ein

Nam

e M

1, g

mol

' pH

Io

nic

stre

ngth

T

emp.

, °C

A

ssoc

n. ty

pe

K0,

dm

3 m

o11

Tab

le 8

. Equ

ilibr

ium

cons

tant

s K0

and a

ssoc

iatio

n num

bers

r of

pro

tein

s in

wat

er so

lutio

ns. T

he ty

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Kq and K,jq• Model calculations should thus be based on 'reasonable'values, i.e. those resembling data from experiments, it is noteworthy thatonly very few equilibrium constants of association have been reported inliterature. Those for polymers are collected in Table 6, those for detergentsin Table 7 and those for proteins in Table 8. Based on these tables, equilibriumconstants of Kq = 2500 dm3 mol and K0 = (K1q)h/(rl -1) 3464dm3 mol' with Kriq = 5050 (dm3 mol 1)24 and association numbers ofq = 2 and r = 50 seem to be reasonable for model calculations. The normal-ized (Mg)a f(c) curves do indeed show the two-step character, thetransition at the critical micelle concentration being less pronounced ascompared with a one-step closed association (Figure 8).

aaa

Figure 8. Concentration dependence of the reduced inverse apparent g-average molecular weightsin the theta state for a molecule-based closed two-step association with q = 2, r = 50, Kq =2500dm3 mol1 and Kriq = 55O (dm3 mol 1)24 (broken lines); the latter equilibrium constantcorresponds to an apparent open equilibrium constant for the addition on a dimer of K034,64 dm3 mol . For comparison, the functions for a corresponding molecule-based closedoiiè-step association with r = 50 and = (KqY('Kriq) = Kr are given (solid lines); thiscorresponds to an open equilibrium constant of K0 = 2933 dm3 mol for the addition of one

unimer. (M1) = (M1),,, = (M1) = 300 dm3 mol 1

An equilibrium constant of Kriq = 5O° (dm3 mol 1)24certainly is in themiddle range of possible equilibrium constants (see Tables 6 and 7). However,model calculations with an increased equilibrium constant of K,jq = s 000°and the same values for the other constants have shown that the two-stepcharacter of these curves will not be apparent within 'experimental' error.Consequently, premicellar equilibria can be observed only if the two equi-librium constants are of the same magnitude. Furthermore, the two associa-tion numbers must be sufficiently different because model calculations withq = 10, r = 50, Kq = 52 dm3 mol and Kr = 5050 (dm3 mol 1)24resulted also in a loss of the two-step character of these curves.

142

C. g cm3


Judging from these calculations, premicellar equilibria would seem tobe detectable only in rare cases. However, many workers in the detergentfield claim the existence of premicellar equilibria1 18125W Most of theirclaims are based on the so-called Debye plots. Debye'26' 127 assumesbasically that a unimer/r-mer system can be treated as a solution of r-mersin a solvent composed of the unimer and the dissolving liquid, i.e. the solutionis treated as a quasibinary system. The method was originally developed forlight scattering measurements, but can obviously be transferred to othermethods, too.

The principle of the method can be demonstrated by, e.g., light scatteringmeasurements. The Rayleigh ratio R is first plotted against the concentra-tion c in order to determine the 'critical micelle concentration' (cmc) andthe ratio Rcmc at the cmc (see Figure 9). Instead of R, the reduced quantityR/K can be used, where K is the optical constant of the light scatteringequations. Following Debye's assumptions, an inverse apparent r-merparticle weight is then plotted against the 'true' r-mer concentration(c — cmc):

1 K(c —cmc) 1=— R + 2(i42)r(C — cmc) (80)

k rJapp cmc k nw

(see also Figure 10). The intercept is supposed to give the weight averageparticle weight of the r-mer and the slope the second virial coefficient of ther-mer/solvent system. An upward bend at lower concentrations has beentraditionally explained as due to the existence of premicellar 18125

because it corresponds to an increase of particle weights lower than Mr withdecreasing concentrations if Debye's method is correct.

c,g cm3

Figure 9. Concentration dependence of the normalized reduced Rayleigh ratio (R/KXM1);1for one-step (I) and two-step (II) molecule-based closed associations. Same assumptions as inFigure 8. 'True' critical micelle concentrations cmc are given by the intersection of the brokenlines with the c-axis. Arrows indicate the 'apparent critical micelle concentrations' cmc' which

were used for the calculations leading to Figure 10. cmc' = (1 ± 0.1) cmc

143

00.1 0.2 0.3 O.L

(-IEU

0

HANS-GEORG ELIAS

c,g cm3

Figure 10. Normalized Debye plot for the determination of association numbers and thedetection of premicellar equilibria. Solid lines indicate two-step associations, broken linesone-step associations. Same assumptions as in Figures 8 and 9. Calculations for two critical

micelle concentrations cmc' assuming Rcmc = 0. Dotted line corresponds to (Mr)w

As has been pointed out previously9, however, the choice of the value ofcmc is very critical. If cmc and Rcmc values left of the asymptote line inFigure 9 are chosen, an upswing is observed even for one-step closed associa-tion (Figure 10), which may be falsely interpreted as due to premicellarequilibria. If the cmc and Rcmc values are chosen right from the asymptoteline, a downward bend appears. We can thus conclude that an upwardbend per se does not necessarily point towards the existence of premicellarequilibria.

But even if premicellar equilibria exist, they cannot be detected by theDebye plot. Again, upward and downward bends are observed, dependingon the choice of cmc and Rcmc The curves for two-step equilibria are onlyquantitatively different from the curves for the one-step equilibria. It iseasy to see why the two-step character cannot show up in a Debye plot:the determination and the choice of a cmc via an R = f(c) plot wipes out alldata points for premicellar equilibria because they must be by necessity atconcentrations lower than the cmc and would thus lead to the physicallyunreasonable negative particle weights in the Debye treatment. Further-more premicellar equilibria are very difficult to detect in plots like Figure 9because they give very small R/K or R values at low concentrations wherethey are most difficult to detect.

We can thus conclude that the Debye plot cannot be used to detectpremicellar equilibria because the very assumptions underlying Debye'streatment preclude the detection of premicellar equilibria even if they exist.On the other hand, the apparent existence of premicellar equilibria as shownby an upward bend in a Debye plot is most likely to result from the particular

144

cmc'

0.255

0 0.3 0./.


choice of cmc and Rcmc Premicellar equilibria, if there are any, wouldprobably be best detected by a plot of the inverse apparent molecular weightagainst the concentration (see Figure 8). Even then, they may not be apparentif the association numbers q and r are too near to each other or if the equi-librium constants 'tKq and 8Kr are very different. The existence of premicellarequilibria thus has not been proven experimentally in a convincing mannerfor closed associations.

ACKNOWLEDGEMENTS

The author appreciates the help of Dr Karel olc, MMI, with the computercalculations and many discussions with him. The help of Professor E. T.Adams, Texas A & M University, in searching data for protein associationis gratefully acknowledged.

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polymolecularity and polydispersity in molecular …

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