The interaction of non-ionic surface-active agents with water

(2)

Phase Behaviour

The phase behaviour of non-ionic surface-active agents in water is typified by the condensed binary phase diagram of jr-dodecylhexaoxyethylene glycol monoether (ClgH~~(OCH&Hg)60H; C&6) (Fig. 1)s. In very dilute solution, the Advan.

Colloid Iwhrfacc Sci., 2 (1969) 297-330

L:~M-LLgiven l~ctnwbgous swies, an iricrcase in the alkyl4~~in length decreases tile c.1n.c. anti increases tlw extent of the niescm~rwphie pltust: regions. For ex~~uplc, at 2S”C, C&J fcmns ;Lhcr~uo#euerais isotropic solution at ill1 compositions, while the Cl&e Lmd (:&(J ~ll~rw~h~~UeS f(Hw lme atid t\viJ cktk.~ IllesOtit(Jr~dtk p~tikX?S respfxtivelyY~4. The hwcr er~nsolute temperature decreases with increasing clkain length. At a particulrrr chin length, the pli~se diagpnls for a nutnhr of different head gmups show broncil~ similar features. Tile tnort: polar head firoups are aswciated with more extensive mesomorphic phase ref;ions, pahcularlp with regard to temperature, higher concentrsrtions for n&x&e hmnation anci the absence of consolute behaviour at normul temperatures. ‘These effects are pa-titularly well illustrated by dodecyldimetl~yl phosphine oxide5 (C~pH25 P(CH3)L~O; Cl31’0) and chclecyldimethyl amine oxide6 (CpzH25 - N(CkI&+O; Cl&O) in which the head groups differ only in the polarity of the co-ordinate bond to the oxygen atom. The l

Advun.

Colloid

XMerjace

Sci.,

2 (1969)

297-330

I’ll

\-I.

I:1

II

\\‘I4

41 ‘1:

304

J. 31. CORKILL,

showala.

These

structures

SOS-IONIC

J. P. GOODMAN,

clearly

have

the same

SURFACE-ACTIVE

AGENTS

q~mmetries as those indiffraction indicates that the

general

dicated by the X-ray diffraction patterns. S-ray chemical nature of the solute tioes not affect tile overall type

of mesomorphic

phase.

The

results

of electron

structure of a particular nlicroscopy on unsaturated

ionic systenls may be taken, therefore, to be representative for non-ionic sy&eins. It must. howcvcr, always be borne in mind that the structures observed in the electron microscope ~nay reflect, in part, the physical and chemical changes brought about by the preparative process rather than the undisturbed structure of the original sample. A pictorial representation of the different aggregation states that may occur as the concentration is increased, from thtlt of molecularly dispersed solute to tllat in Fig. 5.

V

MOLECULES

MIDDLE

SPHERICAL

PHASE

VISCOUS

CYLINDRICAL

MICELLES

ISOTROPIC PHASE

NEAT PHASE

‘-Fig. 5. Schematic representation of idealised struct;lres that centration of surface-active agent increases in a surface-active

(2.3)

hfQlBlZ&

SiRUCTURES

FOR

MESO3IORPHIC

The main structural information

MICELLES

may be encountered as the agent f water system.

cnn-

PHASES

concerning liquid crystalline phases has general types of S-my diffraction

been obtained by low-angle X-ray diffraction.The .-

Advnn.

Colloid

Iuterface

Sci.,

2 (1969)

297-330

PHASE

305

REHAVIOUIt

The observed intensity of the diffracted S-ray beam in a particular direction relative to the incident beam is determined by bottt the 5ynimetry and the climensions of the lattice arrangement of the colIoicI:~l aggregates. The intensity (I(S)) is given

hy nn expression

I(S) = WI where

of the type:

(2. I)

JW.fw>

S is a scattering

variable

relatrcl

to

tile

wnvelengtb

(2)

and

tlte

angle

of

20 bv# S = (2.7 sin 0)/i.. The fnc-tar C(S) includes rrorrectiwi terms for the pol~cr~stalline nature of tile specimen, tlw Lcwentz itlld polark~tion factors, and the instrumenta ret;prxwe ftmction. J(S) is the intcrfercnce function which is determined by the lwrioclicitir2; betivettn tiw structural units and, for large domains, diffraction

it is identical

rd. =

to the l3ragg diffraction

conditiran

that,

for j(S)

to be finite:

2iZ sin 0.

(2.2)

where it is an integer and rt the lattice perioclicity. TJlc form factors, j(S), for various idealisecl particle shapes (rods. 5phert5, sheets) have been calculated~o. \Vith middle (hexagonal) and neat (Iameltar) ljhases. there is a very rapid decrease in intensities as the order of diffraction (11) increases. This change is consistent with the characteristic dimensions of the structural units being an order of magnitude greater than the waveIength of the 4Y-rays (1.5 x). The lack of accurately measurable higher order intensities renders the discrimination between models having the same general symmetries of the structural units extremely diEicult. For example. the observed intensities of diffraction (equatorial) for the middle phase of CIII3I _t Hz0 are in equally good agreement for a model structure of uniform rods (17.9 A) or one of a ‘string of beads’, having the same mean volume per unit length (spheres of the precise dimensions and finer details of 31.0 w radius) It_ The determination the structural units by current S-ray diffraction methods is thus somewhat uncertain. as quite distinct model structures have rather similar intensity patterns. The viscous isotropic (cubic) phase gives an S-ray diffraction pattern with several orders of diffraction that can be indexed on the basis of a face-centered cubic lattice.

There

is no general

agreement

as to the nature

of the structural

unit

in this phase, two main types of unit having been proposed. The first is an assembly of spherical or other colloidal units of the appropriate lattice symmetry, placed at Advun.

Colioid

Imlerfuce

Sci.,

2 (1969)

297-330

306 the

J. M. CORKILL,

XOlc’-IOXIC

SURFACE-ACTi\%

-4GBNTS

lattice points in 5~continuum of solvent l*.lz, the second an inverse of.tliis struc-

ture in which active

the solvent

aggregates

agent the continuum

the ClIIM that

J_ P. GOODMAN,

the

+

Iis0

system,

phase

provide

the structural

*a. The temperature at such a composition

sequence

middle--t

viscous

raising the temperature, shows no abrupt

units and the surface-

dependence

of the conductivity

(GS o/0 w/w surface-active

isotropic

i

neat

discontinuities

iLi

of

agent)

encountered

in passing

on

from middle

phase to viscous isotropic phase *. This situation suggests tlrat, in this system at least, the continuous phase is the solvent in both tile niiddle -and viscous isotropic phases. The combination composition colloidal

of repeat

and density

may

upon the density

example,

both the density

in dilute

micellar

the

nlodel

Sonic

COMJOStTK)N (WEIGHT-I

data

with

the sample

of the dimensions

of the structural

to he assigned

units. There

to the structural

1 have been employed.

typical

results

C,,H21 Cdl

are

nforeover,

contributions showr~

Luzzati

are shown

loid units assumed

unit.

For

to estinxate

in Kg.

Gil,

where

cl &.*a have the densitg the

of

principal

(C&l,),)

Fig. 6. S-ray spacings and structural unit dimensions for C1lII~&OS-S+(CHJ)~ “‘Q°C S-ray long spacings. (0) Structural unit dimensions, X5: Seat phase. -. (0) thickness. V: Viscous isotropic phase. sphere radius, %I: Middle phase, cylinder Isotropic solution. (From ref. 11, by permission of The 12oyal Society.)

X-ray spacings

of the

of the pure liquid solute11 and the part ial speci tic volume

solution’

alkyl chain and head group

units.

the S-ray

assumin, cr a value for the density

particles,

is no general agreement

used separate

spacing?; from

he used in the calculation

together

to have a density

with the dimensions equal

+

H&

half

radius.

of the corresponding

to that of the liquid solute.

at

sheet

S:

col-

It will be

seen that the dependence of the X-ray

long spacing upon composition is much larger than that of the characteristic~dimension of the structural unit within a A dvai.

Colloid

Z&rface

Sci.,

2 (1969) 297-330

particular phase. This phenmnenon is gentxal. Thus, it appear3 that the solvent added to a particular phn~e passes principal\\- into the renyions between the structural unit5 and resulta structural

units

in :m incrca_sr in their

remain

strictly

unaltered,

It can be shown

wparation. then

in a particular

tlint if the

phase: (3.3)

wllere cl, i5 tile esperimrntally of the solute neat, It

and p is an intqer

inicldle

llas been

obeyed,

and viscous

isotropic

n~~opl~nses,

frmncl that

the ideal

relationship5

do not,

in tlw structural

units

indicating

that

The simple nioq>llic: neat

fraction

invcrlvecl

be -1,

--$

in general. alli

ant1 tire vi9~iu5

structure

isotropic

For the nnd -_3.

appear

take

place.

in the tran5itirm

froiri

to be

one

mew-

between

tlie

c-;LIIbe vnvi5:rge:‘l it5 tlw converkm

leatlt~ts tc> tile i\lKI . . . layer

adopted

r;rrluewx

by 5pfwrical

along

tile [ 111: ctirection in face-centered cubic structures. Tllu3, the cfuol c&fthe x*iscous isc~tropic ph:Lse. of tht: neat ph;i_;e btt~onws the ~flll spacing

spcing

If we regard volume

tile spheres

fraction

presents

the

in the viwtnis

for the spherical

n~asiinum

e.g. C11131 f

occurs

to this valuell_

is close 1:rom

uniform

the

point

rods gives

considerably

a possible

slight

viscous

feature

sion of the colloid considerable

that

void

viscous

( IOiO)

unit

is required, in radius

explanation

is based upon the fact the radius

distortion.

at which

the transition

hexagonal

phase.

of spherical

beads

structure

phase.

be formed

which

is that no change while

that

of the rods may

than a uniform isotropic

rod, a

phasell. of densest

this direction

and

it ltas been

(Fig_

the colloid be considerably

derived.

in the characteristic

for the tran4tion

Advan.

is

the

xvi11 be close to that of the (1 1 1)

from

is involved

which

ph*a.se can arise from

of viscous along

of

the middle

phase adhere along the direction will

In

close-packing

If, however,

to uCddle

re-

witlwut

of 0.01 for the middle

isotropic

structure

of this mechanism

reduction

the

of this arrangement

isotropic

fraction

isotropic

cubic

aggregates

water.

Thus,

volume.

fraction

isotropic

can occur

volume

for viscous

from

linear

spacing

of the

of

critical

tlkrn tile

is 0.71, and this concentration

structure

the solute

of the face-centered then

characteristic attractive

view

of the viscous

[li0],

spacing

H&.

phxse ;LS touching,

units

unit is more akin to a string

distortion

packing

than

transition

If the spheres

this

a limitin .q volume

higher

phase structural

of

isotropic

collt)id

at whicll

some systems,

then

feature_;

b will ideally

‘I’ltr transition to ;tnc)tlir:r xvi11nt,\v tw ~~4~:rir;idt:recl.

of the bimA3xilnr units

changes

grcbmetric-al

pli:w~! tylx

phase

long spacing, ‘ipSis the \*olunie g up~w tile tyPe of espansinn?-2. depczndin

cieterminrd

6).

to the ‘smooth

However,

unit

must

in excess

The

dimen-

rod a

an alternative

contain

hydration

of that calculated

Cclloid Zbtterface Sri., 2 (1969) 297-330

30s

J. bi. CORKLLL,

J_ P. GOODMAS,

SOS-IONIC

SURFACE-ACTIVE

AGESTS

from the assumption that its density is the same as that of the pure liquid solute. Since the volume fractions calculated for the transitions on void volume considerations strictly refer to the hydrated unit,it is possible that the iiigllerconcentration limit for the existence of middle pirase occurs when the volume fraction of ‘free’, *as opposed to hydrated, water is 0.09. If this model is adopted, then the viscous isotropic + middle transition may be considered to he due to hydrational changes of the structural unit”r. (2.5) UOLECULAR

STATE

OF

ACClt~GATES

Within the structural units of tile various normal mesomorphic phases there is good evidence that the hydrocarbon ch:Gns forming the interiors have an essentially liquid-like nature. There exists a broad, cliffuae diffraction halo corresponding to a a.5 A spacing, analr~g& to that given by the liquid nlkanesra. Additional direct evidence for this liqui;d-like behaviour has been furnished by density studies. The density of the mesomorphic phase, calculated from the densities of the liquid solute and the solvent, are ohlp about 1 “/, greater than the experimentally determined values; the discrepancy has been attributed to solvent striction upon hydration of the head grtrupll. The application of nuclear magnetic resonance (X3IlP) techniques to mesomorphic phase also indicates that the hydrocarbon is in a Iiquid-like state. The line widths of the alkyl chain proton resonance for the Crs_40 + DzO system decrease from 10 gauss to 0.1 gauss in pnGng from the crystalline solute to either neat or middle phases_ In the micellar solutions and in viscous isotropic phase, a further drop to the order of 0.001 gauss is observed, which is of the order found for liquid hydrocarbon$ a. The line width of tire NSIR signal is determined by the average distance between the magnetic nuclei (the protons) and the relative rate of re-orientation between a given set of nuclei; the higher the re-orientation rate the narrower the line. From a detailed study of the resonance line shapes for middle and neat phases, it has heen concluded that the different segments of the alkyl chain have different characteristic re-orientation rate+. The esperimental evidence for the state of the core in middle and neat systems is thus remarkably consistent and points to a liquid-like structure in which the motions are somewhat more restricted than in the free liquid state. The large decrease in line width in passing from the broad lines of the adjacent middle and neat phases to the viscous isotropic phase is surprising_ The increase in the proton re-orientation rate ha6 been interpreted as due to the rotational diffusion of the spherical colloid units & a whole making a contribution to the re-orientation of the alkyl chain segments in the cares-2. Although such an additional component to the normal thermal motion of the chains would indeed increase the re-orientation rate, the extent of the increase seems too small when examined in the context of the ‘rotating solid’ theory of Kessemeier and Norburg”5. The segmental motion of Advan.

Colloid

Interface

Sci..

2 (1969) 297-330

be deternlined

from surfwe

mea ;tvailable

to restricted, chain

either

per head group

rotation.

tllen

w0u1cl be erpxted_

is a considerable ruiddte

rods

S-ray

and

I\Ieasurenients

liead

required

are

Considerrtbie

systems

has heen

studied

by

derived

when

D&

is substituted

zero

average

isotropic that

in

middle

The

and neat

adjacent

of tlie

relasation

for water.

snlall

l’hases

tenilxraturez6.

Tlie

to alhw

has

times

water Tlie

The

coulAings

(U)

neat

tinws

by ClzE6

units

-& II-,0

or

to crmsist

of

;L solvent

sne2xmmrphic

pltase

of the sc)lvent coupling +

I)-,0

and

in the

n~itlrllrt @k;ses

the

there

neat

in the

sotut ion of

that

nuclei

‘in the CleAO

ant1

(2.1)

the alkpl

in

of in:qpetic

mobilitv

head

the

either

is nssunwi

in tirese

in micekr

in the

relaxation

indicate

or cluadrupoIe

spectra

of

from

collr~icI:rl

tlte

phase,

free. as opposed

motion

in posing

nwasurements

wupling

~AIWAX fornwd

isotrcqk

of

NM R studies.

time

tliermal

phase

of

isotropic

in the

if tile n~itltllc phase

state

nwtion

nw2asurenlents arm

l~y~x~thesis.

cpxhqmle

~~lka.se awl ice)?-l.

front

relaxation

show

density

this tile

tile

siuflkiently

of the

into

by

to tile viscous

g chrmge

group

isc>tropic

to test

i&gilt

either

and phase

\G;cnns

continuunil~~~~, groul3

inweases

in head

iwtropic

the

estent

or neat

a corresponclin

eslxtnsion

to the viscous

smoc,th

to sonle

middle

\Vater

:tre 0)ntinnous

(-

lwoton3

can

lx

effects systems

viscous l/ 100

of

in the

wit11 tltc35e in the

in l;ig. 7. wlkcre tlkcv are slkcwn ;Lli iunc‘timis

:IS indicated bellax-iour

of

tile

cl~cir~ivr~l shift

of

tire water

~xc)tons

Ji

I

-

i

”

1

L

t

I

3.0 RECIPRCCAL

*

3.1 FEMPERATURE

l&

,

(‘Id)

I

34

Fig. 7. Proton magnetic resonance thermal relaxation times (7‘1) for Clo,Ho5(0CHgCHy)&H + H=O as a function af reciprocal temperature. (0) \Vater (de-oxygenated); ( 0) 45 wt-U,: noIute (Isotropic. middle phase); (0) 73 wt-5: solute (Isotropic. neat phase). (From ref. 26, by permission of The Faraday Society.) ,-fdvat~.

Colloid Interf.ce

Sci.,

2 (1969)

297-330

J, Bf. COltKILt.,

310

is

more complex,

on increasing stroyed

in shift occurring

the temperature.

althougli

ikaviour

of

associated

the

with

solvent

with

SUHFACB-ACTIVE

AGENTS

a5 a pfiase boundary

is crossed

This dnrnge has been sIrown to arise from the long orientaticnl is dein the sanrple; if tlie &main

the shift diwontinuity

therefore,

(3)

CLdiscontinuity

orderin, u of the domains

range

KOS-IOSIC

J. F. GOODMAN,

respect

tn composition

molecules,

the formation

small 26_ In tllcse

negligibly

becornes there

there

appear

to

two sv&ems,

is a vru-iation

be

no

cjf the li~luicl crystalline

large

in the

be-

effects

specific:

state.

Alicellar Solutions

(3.1) The been

most

populzw

from tlie optically tuations be

technique

the measurement

related

by

from

calculated

(c) and the osmotic to the intensity R’

where

8

of statistical

is that

pressure

scattering

of the incident

tliermo-

the optical

concentration

to tire solute

at au angle 0 to the lximary

heam,

and

efficiency

solutions,

Uar can he expressed

the angular

beam.

relative

as

m 1110,

the

part iele

is small

destructive

phase

are not randomly

polarised

that

change P(0)

interference intensity

light is completely

dependence

and

from different

of

incident the

scattered

between

light

to unity

will occur other

which than

interference

particles,

The

scattered

will

is used. The light

with

and

function

arises

from different For

in a relative take

two contributions

for 0 =

the wavelength

In systems also

constant,

K,

For the majority 90”.

which can he calculated

at all angles.

will result zero.

The

plane polarised

light

are sma31 compared

is equal

at angles ordered,

tempersiture. in composition.

factor must he employed”0,

when

For particles

effects==.

T the absolute

of the fluctuations

the scattered

from the depolarisation describes

be

light

the ttx~esis scattering,

relates

svstem,

can

eurployed

(3.1)

If it is not, then an additional

scattered

generally

can

=

of micellar

scattering

which

flUC-

(7~). intensity

R is the gas constant

describes

to Deby&9

microscopic

optical

most

ha2i

arises

of tire fluctuations

to the rnncroxcq3ic . ctnkzquences

mechanics ;n-nl their

in a t\vo-component

td

solnti~nw

scattering

Liglit

magnitude

tlleor_v2g. ‘rile

due

micellnr

light.

of a sotuticbn due The

of the systemz7.2*,

tlrat of the solvent The excess

nature

electro-magnetic

equation

dilute

of scattered

and composition.

the methods

parameters

scattering above

inhomogeneous

in loc;rl density

dynamic

for in\~estig!ating

of the intensity

twu

of light,

of the

particles,

decrease

in which

place

from

regions larger

Y(O)

in the

the particles

between

the light

to P(0) are thus exact

A dvak. Colloid IHttrface Sri,, 2- (1969) 297-330

of tl1e fwus tliat

tlw

I’(O)

ulay

(J’(S))

:md lattice

refrac-tive

inclines

be calculated

(J(S)) of

for

the

factors

in S-ray

partickS

and

various

p:irti(-lt*

diffraction

~;(h*ent

theory’.

cl0 not

g:comctrks

differ

i,_\t the

of

Me

(3.3)

and

JI

is the

related

molerular

to the activity

tering

weight

is frequently

expreSsed

1’1 1 90 by a muItildicative assume that each specie3

it may be shown

where

cl and

species. eqn.

Light

(3.2)

from

M,

are

eqn.

the

of

as c -+ 0, SW I’.

The until

excess

the critical

the increase

may

turbidities miceIte

in Bw

that

concantratiun data

intercept

and

in terms

(3.1)

Kc/P llg0

quantity

ii

are

of

the

(virial

be

Jf

is the weight

and

generally obtained

of aqueous

turbidity

(T)

~ot’ffi&nt~)

the excess

whicli

Scat-

is related

tcp

16 ,x/3. For poiyrlispersc ?rystem5, if w13 refractive increment. for dilute solution5

molecular presented

as a function

concentration

as micelles

11 arcs constants

elf the ~~bfuttt. l:rw 5~11 wlutic~ns,

constant of ixk4 the same

scattering

as plots

and

coet1icient

of and,

solutions takes

Rduan.

from

rnolccular

wt-ight in the

of form

the

the

initial

after

which

place

(Fig.

CoCloid Imterfacc

ith

bg the

gradient, agents

Sci..

sohte

c. From

a rapid 8)sS.

weight,

SuggeSted

concentration

of surface-active

is reached.

are formed,

the

average

the

are s~nall

rise, due to

The suggestion 2 (1969)

297-330

312

J. hf. CORKILL,

J. F. GOODSfAN,

NOS-IONIC

SURFACE-ACTIVE

_qGESTS

by Debye” *, that above the critical concentration for micelle formation (co) the scattering is dominated by the micelk, leas been generallg adopted. Thus, the rnicelles are considered to be in zx ‘sokent of the nmnonler at a concentration c,,

made

and the concentration

of xnieelles cl is given

by the tot31 solute concentration

less

CO-

0

0.12

Fig. R. Light I-f&l at 27’C.

024 036 CONCENfRATtON

CM8 x lo* (g

ml-‘)

0.60

scattering turbiclity as a fuuction (From

of concentration of Journal of i’hysieal

ref. 35. bp permission

for C1~IIs&(Cf-13)9, Chemistry.)

3 0

i_

.

2.0.

LO.

Fig. 9. Light scattering Debye plots as a function of micelIar concentration. (0) Type 1 system ClsHssN(CHs)s + 0 + He0 at 27% (From ref. 38. by permission of. Journal of Physical Chemistry.) (0) Type If system C~HJYSO(CH~)~OH +- Hz0 at 22X (From ref. 43, by permission of The Faraday Soeiety.) Advart.

Colloid

Interface

Sci.,

2 (1969)

297-330

MXCELLhR

SOLUTIOSS

313

The results for non-ionic systems fall into two classes: those in which h’cl/IZ$~ is either independent of concentration or shows a slight rise (Type I) and systems in which I
*

Of the several methods of molecular weight determination that are available using analytical ultracentrifugation, those that are independent of micelk shape depend upon equilibrium being e~tablishecl between the sedimentation and cliffusion processes in the cell. It has been shown by Arcl3xdd 46 that the approach to sedimentation equilibrium can also be rigorously analysed. As this Inethocl is less time-consuming than full equilibration. it is increasing in popularitg. Both of these equilibrium methods lead to an apparent solute tnolecular weight which must he . corrected, in exactly the same way as light scattering data, for solution nonideality effects_ In the C&a system, the increase in apparent micelle molecular weight with increasing temperature, as determined by light scattering*?, is in good agreement with the results obtained from the Archibald methodA”_ In contrast to other types of macromolecular systems, the direct measurement of osmotic pressure of solutions of surface-active agents has not been possible due to the difficulty in preparing a membrane which is non-permeable to both monomer and micelle. However, the effective molal concentration of a solution may be determined by vapour pressure osmometry in which the difference in temperature between drops of solution and solvent, equilibrated through the sapour phase, Advan.

Colloid Iderface

Sci., CL(1969)

297-330

314

J. %I. CORKILL,

J. F. GOODMAX,

SON-IOSIC

SURFACE-ACTIVE

AGENTS

is measured using thermistor beads 18. An example of the use of this method for the determination of low molecular weights is discussd in Section 1. (3.2) SIZE ASL, SHAPE OP

5fICELLIcS

In the study of ~nacronlolecul:tr systems, inlormaticm 011 tile size and k~pr* of the solute moIecules can be obtained from tllc l~ydrod\-natnic tIYlIlSp0rt propcrtii5 _ In view of the estensive and fruitful use of ~;edimentation, diffusicm and vkcci.;ity measurements in the investignticn~ of protein 5olution49, it ia wrprisiq to find that very few micelIar systems ltn~w twen examined 1)~ these tecImiclut5;. l;or ideal, dilute solutions. the setlimentation (SO) ant1 tliffu.;ion (I)“) cc~tA%:ittnti;are wlntrd to the hydrodynamic frictional coefficient /, 1)~ : (X-l)

(3.5)

where l and Z,:!are the solute molecutar wcigtit and partial qxilic volunir, t) the soixwlt clewity and N, k, T the _-\vogadro nunlber, iIoItzmann constant ant1 u&wlute temperature respectively. The factor / is the cottfticient of proportionality between the viscous drag upon tile particle and tire rtktive velocity between the particle and the medium at large distances from the particle. It is related both to thr shape and volume of the hydrateti particle and was lirs;t deduced, by Stoke.9 tch be given by: i / ”

62zqN,

(3.6)

.

for spheres of radius R in a medium of viscoAty q. His treatment was later estended by Perrin 51 to ellipsoids of revolution (randomly orientated). Tiw results of these calculations are usually expressed as a function of the axial ratio of the ellipsoid in terms of the ratio of the particle friction factor to that for a sphere (JO) of the same volume. The intrinsic viscol;ity pq] of a macromolecular species is also related to the shape and hydration of the particle, and can be written in the form: 1173=

/1(-p)

(3.7)

Efh

where A(P) is a shape factor and oh is the volume of the hydrated kinetic unit per gram of solute. For spheric&l solute particles, A(#) has the value 2.5 and for randomly orientated ellipsoids of revolution, SirnhasL has obtained A($) as a function of the axial ratio. Roth the intrinsic viscosity and the frictional coefIicient are related to the extent of hydration of the solute species, l&t are affected to Advan.

Colloid Interface

Sci.,

2 (1969)

297-330

0

OS

MICELLAR

l.0

15

CONCENTRATION

2.0

I 10” fg ml-‘)

Fig. 10. &Iicelle molecular wrights as a function of micellar concentration for C&II~SO(CI-I&OH t_ D-0 at 22’C. (0) Light scattering data, ideal solution approximation. (a) Sedimentation and &codty data using Scheraga and bfandrlkern35 treatment. d&an.

Cdioid

Interftice

Sci..

2 (1969)

%7-330

316

J. M. CORKLLL,

J. F.

COODJIAN,

NON-IOSIC

SURFACE-ACTIVE

AGENTS

actions to the sedimentation constant may, in dilute solutions, beestimatedby the theory of Burgers55 as modified by Pyun and Fixman 56. Similarly, viscosity data may be corrected by the use of the semi-empirical relationship of Hug&&‘. In one Type

II system,

C&&

+

DpO, the molecular

weights

as a function

of concentra-

tion derived from transport properties, using the Scheraga-Mandelkern treatment, are in good agreement with those obtained from light scattering, assuming solution ideality (Pig. 10)5”. Although in other systems c>f Type II the light scattering behaviour

may be attrihutecl,

at least in part, to activity

effects,

in this system

the

of evidence is in favour of the rnicelle growth model. As we shall see later, an increase in micelle size with concentration is to be generally expected for q-5 terus with a distribution of micclle sizes.

balance

(4)

(-!.I)

Thermodynamic

MODELS

FOR

Properties

blICELLIZATiO3

. The thermodynamic clescription of the micellar state leas, in the past, been a kinetic equilibrium between the nwnomer and approached from two viewpoints, a single micellar specie+, and a phase separation 60. It has long been realised that although both these apprortches are adequate in erplainin, CTsome fee tures of uiicelle formation, both are over-simplified descriptions of the micelle state. A general statistical mechanical treatment. in which the micelIes are treated as clusters as in the Mayer theory of imperfect gases, has been made by Aranow6l. The ‘small system’ thermodynamics developed by Hills? have been applied b_v Hall and Pethicaa to obtain some exact relationships between the tl~ermodynamic parameters of a micel!ar system. Hill introduced into the classical thermodynamic equations an extra variable descriptive of the size of the small system and then developed equations in which the thermodynamic properties are themselves functions of a mean size parameter. An equally rigorous approach, which has some -similarities with the treatment of protein aggregation’5=64, is to consider the relationships of a kinetic equilibrium between micelles of different size&s. The intrinsic properties of the individual micellar species may then be removed from the relationships by the appropriate averaging procedures. For our present purposes, we shall employ this approach to obtain relationships between the average micelle size and micellar concentration and the solution colligative properties. We shall also examine the connection between the c.m.c. and the thermodynamic parameters (free energy, enthalpy, entropy) of micelle formation. This treatment is presented in some detail as it provides a comprehensive basis for the description of the solution properties of polydisperse associating systems. Advats. Colloid

Inter/ace Sci.. 2 (1969) 297-330

THERJIODYSh3I1C

We

may

espress

solute

present,

solute

monomer

species

317

PROPERTIES

the

with

to the weight

and solvent

contributes

of the solution

composition

respect a term

in terms

fraction

by a stoichiometric IL~_x~.we have

of the total

xnd the molecular mole

for dilute

fraction

quwtity

weights

xt. As each

of

of the micellar

solutions:

r

.rt = The

colligative

~9cr.x~ +

(-L-6)

.T~.

properties

of the solution

of “molecuIes”

of each of the solute

a colligatively

effective

mole

will be determined

species

fraction,

present,

by the relative

and so we may

numbers

similzwly

define

xc, by:

_-Idvnn.

CoEloid

Itlterfurc

Sci..

2 (1969)

297-330

q J_ Bf. CORKILL,

318

J_ F. GOODMAN,

The number sverage association

NON-IOKIC

SURFACE-ACTIVE

number of the solute

AGEETS

with respect

to all

solute species Nn is given simply by division of eqn. (4.6) by eqn. (4-7):

(4-S)

Since xc is formally related to the -solution colligative motic pressure ;z by: XT nz =

-

FI

%,

(Xv .g

properties, such as the os-

1)‘

-

where Vl is the partial molar volume of the solvent, _$I8 may be determined from the experimental relation between xt and xc from suitable colligntive property determinations such as 1.apour pressure measurematS. By differentiation of eqn. (4.6) and (4-7) at constant sure and substitution

dx,

=

The weight defined by:

ford $R~_v, and cl &,

(C n,.rr

i_ x2) d In x2_

average

aggregation

temper:rture :uld pres-

from eqns_ (4.3) ~ncl (4.4), we obtain:

(4.11)

number SW with respect to all solute species is

(4.12)

and from eqns. (4.10) and (4.11). -.

.iiw=----

we have:

drt d&

(4.13a)

*

N,.may be obtained from vapour pressure measurements by graphical differentiation of the experimentally determined relation between xt and x,, or directly from Iight scattering data using eqns. (3.1) and (4.9). We are justified in employing the .two-component eqn. (3-l), rather than a multi-component treatmentza, since in these systems there is only one independent

compositional

variable.

N, may also be obtained from light scattering data by the following method. By integration of eqn. (4.13a) written in the fork: Advo~. Colioid fsterfucc

Sci., 2 (1969) 297-330

xe may tx determined ;I< 8 function of St ant1 hem-e. from erln. (-IS), z, n-my be calculated. .Alth~mg$~ h,tll liglht scattering anrt vqxbur pressure merr5urements are in principle interchangeable, from the prxtkal point of view theruwelectric osnmmetry pwvides the more nceurate ci;rt;i txlcw- the c.tn.c. and light scattering measurclnlents rtbove it. (4.3)

MICIILLI:

SIZE

r (?I,>

=

xB If#. -t-r

(4.14;

--r--

x sy ancl (4.15)

and the variation of sz with xt may be cabtained from the experimentally determined variation of xr with _rt bbV integration from the limit below the C.IXLC. for the C&T + H&l detined by xt z xe 21 x2. An example of this procedure system is shown in Fig, 11. The values of (11~) and ~ thus obtained as SLfunetion of _rt are shown in Fig. 1267. An equivalent esprasion to eqn. (4.16~~) which is more convenient for the treatment of light scattering (1.10) by division by .t;t and re-arrangement to give:

d In xz =

I

The distribution by the free energy

d In xt NW

from eqn.

(-I_ 16b)

.

of the solute among the various

of formation

data is obtained

of the particular

micellar

species is governed

species and the total solute con-

AJVU~L Cd&d

ltrterfuce

Sci.. 2 (1969)

297-330

320

J. M. CORKILL,

J. F.

GOODMIAN,

NON-IOSIC

Pig. 11. Colligative properties of CaIil?ir;+(CHs)n(C~i9_)JSOsmole fraction (xt) at 25’C. (0) xc. colligativr mole fraction ments. (-) x2, monomer molt: fraction from eqn. (4.16a).

centration.

The relation

between

the micelle distribution

tion may be derived by differentiation

The quantity



SURFACE-ACTIVE

+-

1ItO

from

as a function

vapour

pressure

AGENTS

of

t-da1

measure-

and the sointe concentra-

of eqn. (4.14):

is related to the weight average association

number

w by: (1.19) and we may

therefore

express

eqn.

(1.18)

in the

form:

(4.20) For any distribution, since >/ ~ calculated from the dependence of (nr> upon x2 are shown in Fig. 13, in addition to those calculated directly from xt and xz (eqn. (4.16~~)). A &fan. CoUoid

Interface

Sci.,

Z (1969)

297-330

THERJfOl>TSAMIC

321

PROPERTIES

. -, MoLE

W, - t03

FRACTION1

numtwrs for CBlI IiSt(C)fy)r(C~1=)JS09Fig. 13. Aggregatiou -+ H& as a function of total mok fraction (.rt) at 2S’C. Lower curwz: . numkr awxag:r a ggrcgation numb3 from xp as a function of xt (vapur prr5rure data). Upper curve: - w, weight a\*erage: aggregation uumber hii1 derivative of x, as a function of xt. [ 0) < n,; ,,. calcufated from x-ariation of OL,> with _rr from eqn. (4.‘10).

In defining to two different guishable 0Pr

the standard

q>ecies,

tllen

From

~,pgJ.

free energy of micellizxtion,

micelle standard we

eqn.

are

(4.2)

statt.5.

we may refer the process

If we regard the micelieS as forming

naturally

led to consider

we have,

denoting

ttle average

averages

with

elf the

u-eight

distinterins

factors

.r,

by<>; <(PP

-

If we regard

‘r&q>

= RT(
all micelies

yro, in a standard

(I) micellar

species

types

-

(in

(4.21)

sr>)

as indistinguishab1e

free energy,

from

II1 x;

species,

state corresponding

the ‘isolated’

differ by the appropriate

standard

entropy

then

we may

to the mixture states.

of mixing

These

consider

the

of the individual

two standard

state

term:

(1.22) and hence

the free energy

difference

between

the states

is: (4.23)

From eqns. (-1.21) and (-L-23), after division by CM,.>to obtain monomeric

A&p,

surface-active

with respect

4G9’

agent,

to the mired

we obtain

for the

the average

free energy

per mole of

of miceliization,

state:

ln( E xr) (nr>

1 i Advan.

(1.24) Celloid Xttterface Sci., 3 (1969) 297-330

322

J. M. CORKILL,

J. F. GOOI>X4N,

SOS-IOSXC

SURFACE-.4CTIVE

AGESTS

Eqn. (4.2-k), unlike eqn. (4.21). contains quantities that can be readily determined experimentally from colligative data, and are independent of assurllptions concerning the micelle distribution. The standard enthalpy of miceliization may be obtained in terms of x2 and C sr by differentiation of eqn. (4.2) with respect to temperature at constant presliure. Denoting tile enthalpy mole of micelle) of the uth species by AIf,O, we have:

of micellization

(per

En order to obtain the average value, , we multiply each sic& by xr, sun1 cJ\*er all micelle species (r) and divide 1)y 5.ry . After re-arrangement and division (q.), we obtain the enthaIpy of micellization per mole of mnnomer AHq”:

by

Thk standard entropy of micelle formation (referred to the ‘mixed’ state) may be obtained from eqns. (4.24) and (4.26). are exact for ideal sc~lrlticm ccbnThe espressions derived for A&O, and Ail\Hzo ditions. In principle, with sufficiently accurate colligative solution data, all the terms on the RI-IS. of these equations can be esperimentally determined and the influence of (9~~) upon AC20 and AHp obtained. Tn practice, tlrrxe systems for which accurate data can be obtained invoive rather concentrated solutions and the assumptions concerning solution ideality are likely to be invalid. \Ve are therefore led to consider approximate forms of these equations and in particular their relationship to the c.m.c. The c.m_c. is generally defined as the concentration at which an abrupt change takes place in the derivative of some system property (surface tension, turbidity, density) with respect to concentration. The analysis of the colligative properties of some non-ionic systems with <.ltr> > 20 above the c.m.c. has shown that up to the c.m.c. (xt*). defined experimentally, ;Y~/x~* > 0.9. Above the C.IILC., ~2 remains almost constant. showing an increase of about 1 o/o at five times xt*. cl& There is thus a narrow concentration range in which 2 changes from - 1 to -0. The experimental particular

determination

value for 2

of the c.m.c. is thus*equivalent

to choosing a

and so for different methods of determining

the c.m.c.

narrow ranges of values of xt* are obtained. It can be shown that for systems in which > 20 that the approximations introduced by setting xi equal to the total solute concktration at the c.m.c. and neglecting the term involving the micellar concentration (In !&) introduce a negligible error into the expression for &G$J. In the subsequent discussioti, the free energies of micelle formation have been Advaa.

Colloid

I&face

Sci.. 2 (1969)

297-330

T~lERMO:~YS_~~lIC

323

PROPERTIES

calculated from the experiuwntally approximate formula, A@

E RT in A-~*.

The entlialpy of nCcellizatiou culatecl frotu the teinperaturc

deternlinect

c.ul.c.‘s

(mole fraction

xt*) by the

(4.27) may, to a 5irtiilar dcgrec c~f ;~l’l’rc)sirll.2tit)n, be cdtlepenclcnce of xl* using t t12 relation:

Fig. IS. Standarcl free energies of micellization as a function of chain length at S3’C. from c.m.c. of the Faradav Society.) data. (0) C,HeHir(OC)IyC~I~)mO~I (F ram ref. 68, t_~y permission (0) C,,lI~n+lN(CH&+O (From ref. 38, hy permissmn of The Journal of F’hpsical khemistry.)

J. M. CORKILL,

324

J. F. GOODMAN.

NOX-IONIC

SURFACE-ACTIVE

AGESTS

more negative as the polarity of the head group decreases. For erample,in passing from the C,AO to the C,Es series, a&o decreases by Y 1.8 kcal/mole. AHz* decreases with increasing chain length for both series 6B.70, although the increnrent is marginally smaller than that for A& 0. The entropic (&SSO) of micellization is always positive, but due to the sinlilar incremental priperties of A&O and AHPo, only weakly dependent upon chain length; the trend iS towards increasing values as the chain

length

In general,

increases. ACe*

extrapolates

carbon atoms, leavinglnrge be attributed

pairing12

and conlprnsating

to the solxxtion

groups upon the nlicelle

to a value

clt~~ges

surface

and the neutralisathn

of zero

for

;L chain

heat and entropy

accon~panying

and con~pnred

of the nlipllatir

tile close

wit11 sinlilar

carboxylic

length

of 2-4

ternls72. These packing

tiffects

may

of the l~~cl

for ion The process nf

observetl

widsi3.

micellization is therefore promoteri by the laq;rr cohesive forces between water molecules them those between water and tile =alkyl ch:dti7*. The increment in AGnO is iLmeaware of this difference nncl if we attribute a surface area to tire nwtlry-

he group of 7B 15 As. then we obtain an energy clmnge of ~33 erK/criG, This figure is of the same order of magnitude as the inttxfacinl frtw energy betwwn water

and a bulk hydrocarbon

NUMBER I’lg.

WG

CJI C~GROUPS

plu~s~ (-50

BETWEEN

tc(t.

SO

crglctr12).

AN0

&i

GROUPSbd

14. Thertnociynanilc parameters of miccllitatlo~r fur C,14rN + ~SO(CI-I~)mOiI + (0) CIJHI~, (0) CnHlr. (From ref. 76, by permlssion of ‘I’he Varaduy Society.) Advan.

c!olkoid Infevfsace Std..

!J

IlaO at

(1OCU.I)297-330

The f

investijintic>n

of the tl~ermtxl~narnic

lx4iaxViiour of ~lO~l-il~Jnic surfactaut

Wilter systems has been focused upon tlie dilute

formation

takes

place.

In more

concentrated

solution

scrlutiom,

region

althmglk

in which there

micelle

is consider-

able information on structural aspects of ruescm~cq~liic phase foriuation~~, there is a dearth of thermcidynamic data. In ;Lstud_v t>f the C 1,:I :a + Ha0 $‘ste111, the escess functions of mixing were found to be typical of ;L binary aqueous systeiu in \\+ich strong solvent-solute interxtion takes plnce 3_ The excess free energies are relatively small due to tlie approximate cancelltition of the large negative exess entlralpy and entropy functions (Fig. 16)“. Tl ke excess entlialpy is associated wit11 the hydration of the head group, xvhile the entropy term arises essentially from the large disparity in the nlolrrr vohnes of tile solvent and the associated solute species. A comparison show

different

phase

of three

honwlogues

bellaviour,

gives

(C&J,

II =

8, 10 and

12), which

at 25’C

data that are virtually superbasis’ 3.1. A study of the heat capaci-

experimental

inqmsable when expressed on a mole fraction ties and expansibilities in the Cl&j + Hz0 system has revealed that there arc only very small discontinuities in both the volume and enthalpy on mesophase formation, the changes being two orders of magnitude less than those associated with the melting of the pure solute 3. The co-existence region between isotropic solutions and the mesomorphic pleases is also very narrow (Pig. 17)3, which may be regarded as a necessary consequence of the Gibbs-Konovalow relationship65 for first-order phase transformations in binary systems. Consequently, the siructural Advaa. Colloid Intrrjace Sci., 2 (1969) 297-330

l.OC)-

0.9!j-

3-

5-

a8

0

a05

0.15

0.10 MOLE

Althaugh phase

the factors

are unknown,

a20

FRACTION

that govern

the organis;rtion

C

C&,

the fornmtion of rod-like

of a particuhr

units

Advarr. Colloid

type of nwm-

into an ordered lderfuce

Sci.,

2 (1969)

structure 297-330

Aduan.

Colloid

Inlcrfuce

Sci.,

2 (1969)

297-330