Determination of thermodynamic parameters of polymer-solvent systems by light scattering

Euro~)can Polymer Journal, 1970. Vol. 6, pp. 1063-I074. Pergamon Press. Printed in England.

D E T E R M I N A T I O N OF T H E R M O D Y N A M I C PARAMETERS OF P O L Y M E R - S O L V E N T SYSTEMS BY LIGHT SCATTERING TH. G. SCHOLTE Central Laboratory, Staatsmijnen/DSM, Geleen, The Netherlands (Received 27 October 1969)

Abstract--From the intensities of light scattered from dilute and concentrated solutions of a given polymer, the chemical potential of the polymer and the solvent can be determined. Experiments have been carried out with solutions of monodisperse polystyrenes of different molecular weights in a poor solvent (cyclohexane) and in a good solvent (toluene) at different temperatures and at concentrations up to 30 per cent by weight. For all concentrations, the interaction parameter X appears to be slightly dependent on the molecular weight. The results are compared with those determined by equilibrium ultracentrifugation and critical miscibility measurements. 1. I N T R O D U C T I O N IN THE theory of light scattering of solutions, (t-s) one generally starts with the extension, given by Einstein, C~) of Smoluchowski's (~) fluctuation theory. According to this theory, the scattered light intensity is given by the mean square of the concentration fluctuations in small volume elements, a value determined by the second derivative of the free enthalpy to the concentration. Zimm and Dory (6) found that, in the case of a polydisperse polymer, the weight average molecular weight can be determined from the scattered light intensity. Brinkman and Hermans (7) derived an exact expression for the turbidity of a polydisperse polymer solution, starting from Zernike's (s) scattering theory of multicomponent systems. Soon afterwards, Kirkwood and Goldberg (gJ and also Stockmayer ~l°~ gave a derivation, which is basicalIy identical to that of the above mentioned authors. According to these theories, one finds the turbidity as the sum of two terms, namely the turbidity caused by density fluctuations and the turbidity caused by concentration fluctuations. Because the second term is often more interesting, the first one can be eliminated by subtracting the scattering intensity of the pure solvent from that of the solution. In the vicinity of a critical point, the scattered light intensity can rise to very high vatues/lO, xl> Under these conditions the theory shows deviations, because some factors, such as coupling between the fluctuations at the various places and the influence of extra energy caused by concentration gradients, will join in and make very large fluctuations improbable. Strictly speaking, the above theories are only valid for the Rayleigh scattering, where particles and volume elements are small relative to the wave length of the light. In the case of Debye scattering, however, the scattering intensities extrapolated to zero angle depend in the same way on the concentration fluctuations as in Rayleigh scattering, so that in this case also the above theories can be used. 1063

1064

TH. G. SCHOLTE

In our w o r k we examined how far it is possible to a p p l y these theories to concentrated p o l y m e r solutions (up to 30 p e r cent by weight) a n d to calculate the chemical potentials o f solvent and p o l y m e r for this c o n c e n t r a t i o n range from the experimental data.

2. E X P E R I M E N T A L The light-scattering measurements were performed with a Sofica photogoniometer. The temperature was controlled by means of circulating water from a thermostat. Measuring cells of a special shape were used to get a good temperature homogeneity in the cell (see Fig. 1). The solutions were made dust-free by filtering them through a commercial millipore filter.

~xx-.-.\-, \ ~ \ - .\-...~

FIG. 1. Original and improved measuring cell. Refractive index increments were measured with a self-built differential refractometer, suited for measurements on concentrated solutions.
Mw = 51,000 Mw = 163,000 and M,, = 520,000.

The cyclohexane was spectroscopically pure (spectrosol), obtained from Hopkin & Williams Ltd. The toluene was Baker Analysed Reagent p.a. Both solvents were used without further purification. 3. T H E O R Y F o r the intensity o f the scattered light, caused by the c o n c e n t r a t i o n fluctuations, a n d e x t r a p o l a t e d to zero scattering angle O, the following f o r m u l a is valid: (~AI)0=o - - 2 r, 2 n 2 (An) 2. . . . AV.

(1)

A'* Rn In this e q u a t i o n A I is the scattered light intensity o f the solution minus t h a t o f the pure solvent, relative to the light intensity o f pure benzene scattered p e r p e n d i c u l a r to the incident ray. a is the well k n o w n angle factor (sin 0)/(1 + cos20), nn the refractive index o f benzene, RB the R a y l e i g h f a c t o r o f benzene a n d A the wave length in vacuo o f the light. (An)~onc. is the a v e r a g e o f the square o f the refractive index fluctuations caused by the c o n c e n t r a t i o n fluctuations in small volume elements A K

Determination of Thermodynamic Parameters of Polymer-Solvent Systems

1065

An is related to the concentration fluctuation Ac by the refractive index increment

dn/dc. For a polydisperse polymer solution, (Ac) 2 can be determined by means of statistical thermodynamics, leading to the formula:

(aAI)o=o

- '

"

h'* R~

\dw/

~ 7'

[b I

AV,

(2)

where k and T have their usual meaning, I bl is the determinant with elements & (AG)/Owk 0w,, AG ----free enthalpy (Gibbs free energy) of mixing of a volume element A V, Wkand w~are the concentrations (weight fractions) of polymer components k and l respectively, w = the total polymer concentration (w = Zw~) and B~i is the cofactor of element i, j of the determinant. To determine the elements ~2 (AG)/Owk aw~, we must know how AG depends on the total concentration and the composition of the solution. For this purpose, use can be made of the Flory and Huggins ~ ~ - ~ equation for the free enthalpy of mixing. We shall use this equation with the concentration expressed in weight fractions. ~5~ AG [ N~-'Mo , lpAV ~-~ - - Wo In Wo ÷/----uM'~l w, In w, -v g Wo w - ~ - o '

(3)

i

where: ,AG = O = wo = Mo = Ml = g =

free enthalpy of mixing of pAV gram of solution density of the solution weight fraction of the solvent molecular weight of the solvent molecular weight of component i of the polymer free enthalpy correction parameter.

Differentiation of AG with respect to the numbers of moles of the various components yields the well-known expressions for the chemical potentials: A/x°--In(1 - - w ) + ( 1 - - ~M°) w + [ g - - ( 1 -- w) g-~w cg] w2 RT

(4)

and ,.X~,- I n w , + l

RT

---

M~ M,

w - - - ( 1Mi -w)-r-

Mo

, [g + w

~og]M~

cw]

(l-w)-.

,

(5)

Atzo = (chemical potential of the solvent in solution)-(chemical potential of the pure solvent) A/x, = (chemical potential of polymer component i in solution)-(the same in the pure state) M. = number average molecular weight. The factor [g -- (1 -- w) ~g/~w] is Huggins' interaction parameter X. We shall denote it by X~, to indicate that it belongs to the Flory-Hug~ns equation in which weight fractions are used.

1066

TH. G. SCHOLTE

If we assume that the interaction parameter g depends on concentration and temperature but not on the molecular weights, a twofold differentiation of AG leads to the following expressions for 82 (:~G)/c3w~~wl:

i = y:

a2(AG) -- RTp A V (

i # j:

02(AG) -- RTp A V K 0w~ ~w1 1 I

1

K = M'-'~o 1 -- w

with

1

+ x)

(6) (7)

+02{w(l--w)g}] 3w z

(8)

(see also Refs. 16 and 17). Substitution of these values in the expression for the scattered light intensity yields:

_ (~A/)e=°

2~r 2n~ (dn~21

wM,,

(9)

Na A4 Rs \dw] -p I + Kw M,,"

M,, = weight average molecular weight, Na = Avogadro's number. This equation relates the scattered light intensity to K and hence to the interaction parameter and chemical potentials of the solution. We can derive the relation to the chemical potential of the solvent as follows. Differentiation of A/~o (Eqn. 4) to w, at constant Mn, yields:

I R---T\

8w ]M,

=

- Mo wK-

Mo

(lO)

M"-~"

Substitution of the value of K from Eqn. (9) gives:

-- {O(A/z°)~ = RTMo [ 2rr2 n~ (dnl2 w 1 \ Ow ]M. L~VA-~4~. \dw] p(aAI)o -F Mo

1 ]

I~., "

(11)

The right-hand side of this equation can be determined experimentally and yields O(A~o)/aw at the given concentration. When this has been done for a number of concentrations, from very low values upwards, subsequent integration of O(A/zo)/aw gives the quantity Atzo for all desired values of w. The interaction parameter Xw can now also be determined. In an analogous way, namely by differentiation of A/~, to w at constant molecular weight distribution and summation over all i's, we can derive an expression for the chemical potential (actually the number average chemical potential, see Ref. 15) of the polymer.

(@')

~,wo

=RTMnr 2--~2-~ (dn) 2 1-w

LNA A" Rn

~w

l--w(1 1 )] ~-AI)-o -? w .l~o 1~,,,

(12)

It is interesting to compare equations (11)and (12)with the corresponding ones in the theory of equilibrium ultracentrifugation of polydisperse polymer solutions (equations 22 and 23 of Ref. 15).

Determination of Thermodynamic Parameters of Polymer-Solvent Systems

1067

RESULTS With the polystyrene-cyclohexane solutions, experiments were performed at 35 °, 45 ° and 65 ° at concentrations from ½ per cent upwards. With sample 7a (Mol. wt. 51,000) we could go as high as 30 per cent by weight, with sample la to 20 per cent and with sample 5a to 10 per cent. Use was made of the refractive index increment values :c~5~ 35°: 45°: 65°:

dn/dw ----0.131 dn/dw ----0.133 dn/dw = 0.137

+ 0.070 w -5 0.058 w2 q- 0-070 w -+- 0-058 w: + 0.070 w + 0.060 w2,

and density values :(15~ 35°: p = 0.7642 -k- 0.222 w + 0-023 w2 45°: p = 0.7546 + 0.222 w + 0.03 w2 65°: p -- 0.7355 + 0.222 w + 0.04 w2. The Rayleigh factor of benzene at the wave length used (546 m~) and the calibration temperature (25 °) was Ra = 16.3 × 10 - 6 c m - 1 ; the refractive index of benzene at this wave length and temperature was ns = 1.502. Figure 2 shows ~tz/~was a function of w for polystyrene in cyclohexane. As can be expected, the values are high at first but, instead of constantly decreasing as for

7a

la

0

5m

I

x

f

"6 E .L

t

\_jr

I O

0-I

0.2

0.3

0

0-I

0,2

w

[ZIG. 2.

d~/~w VS. concentration of potystyrene--cyclohexane solutions for the three polyO:

352;

sytrene samples. ~ : 45°;

A:

652 .

•

0"00713 0'00709 0"00701

0'01432 0'01424 0-01408

0"0288 0"0287 0-0284

0"0437 0"9434 0-0429

0"0588 0"0584 0-0577

0"0741 0-0736 0"0727

0"1134 0-1127 0"1112

0"15,13 0"1532 0"1513

0-2405 0"2388 0"2359

t ('C)

35 45 65

35 45 65

35 45 65

35 45 65

35 45 65

35 45 65

35 45 65

35 45 65

35 45 65

(g/g)

0'01

0'02

0'04

0.06

0.08

0-10

0"15

0.20

0"30

IV

49"64 67"38 101"0

15"95 22"33 35"26

9.21 12-51 19"27

5"153 6"535 9-362

3"906 4-796 6-593

2"799 3"326 4-353

8"88 13"98 24"52

4.48 7"32 12"75

2"017 3"351 5"491

1-418 2"286 3-573

0"959 1.453 2"135

0-590 0-817 1.120

0"282 0"344 0"426

0"885 0"962 1-126

1"807 2"055 2"564

0-1,10 0.159 0"180

1a (M. = 163000)

0-440 0-467 0-524

7a (M. = 51000)

1.075 2"289 4-212

0-710 1.431 2-624

0-435 0.822 1-462

0"243 0"408 0-674

0"107 0"148 0.206

0.051 0.063 0"076

5a (M ~ -520000)

--At~o × 10 -6 (crg/mole)

0"6139 0"6071 0"5955

0.5716 0.5660 0"5559

0.5519 0-5459 0"5375

0.5331 0.5286 0-5200

0.5260 0-5215 0-5131

0"5191 0.5147 0.5065

0-5126 0.5082 0"4997

0.5065 0.5017 0.4926

0"5032 0.4983 0"4884

7a ( M . =51000)

0.5726 0-5681 0-5596

0'5522 0-5477 0.5400

0"5337 0.5289 0.5221

0-5266 0'5217 0-5154

0"5196 0"5147 0"5089

0"5130 0"5081 0"5025

0.5065 0.5016 0"4962

0.5034 0.4978 0.4939

la (M. = 163000)

Xw ..............................

0.5335 0.5292 0-5230

0"5263 0"5223 0-5162

0"5192 0-5155 0"5097

0"5123 0"5090 0"5035

0.5059 0.5025 0"4982

0"5027 0.4989 0"4957

5a (M. = 520000)

0.5734 0.5616 0.5405

0.5463 0.5362 0.5169

0.5331 0.5239 0.5057

0-5203 0"5116 0-4946

0.5160 0.5073 0-4908

0-5112 0.5020 0.4857

0.5076 0.4989 0.4817

0.5022 0.4931 0.4741

0.5002 0.4907 0.4701

7it (M~ -51000)

TABLE 1. CItEMICAL POTENTIALS AND IN'FERAC'FION I*ARAMETERS OF POLYSTYRENE-CYCLOIIEXANE SOLU'I IONS

0.5483 0-5399 0.5236

0.5341 0-5257 0"5104

0-5217 0.5126 0"4990

0.5171 0.5078 0-4950

0.5126 0.5032 0.4913

0-5084 0.4988 0"4873

0.5044 0.4945 0.4834

0.5023 0.4914 0.4832

llt (M~ = 163000)

X~

0.5218 0.5134 0.5011

0.5168 0.5091 0.4969

0-5121 0.5048 0-4930

0.5077 0.5007 0.4897

0-5036 0-4967 0.4880

0.5017 0.4942 0.4882

5 a (M+-: 520000)

© ,.q rn

(3

O

Determination of Thermodynamic Parameters of Polymer-Solvent Systems

1069

an ideal or pseudo-ideal solution, increase after having passed through a minimum. There is a strong influence of the temperature and also of the molecular weight. The minima lie at about 12, 8 and 6 per cent by weight for the three polystyrene samples. These values are slightly lower than the critical concentrations, 14.6, 9.9 and 6-4 per cent by weight, respectively, determined by the phase volume ratio method. (ls~ From ~(:_kffo)/~w, also calculated from the experimental data, ~ffo has been determined by integration. The interaction parameter x, in its turn, has been derived from -X~z0.Table 1 shows ±/zo and Xw for the three samples and the three different temperatures. In addition the volume fractions 8 and the interaction parameter x,, from the Flory and Huggins equation, expressed in volume fractions are given in this table.

0"6I 45 ° 65°

0

! 0.1

T 0-2

FIG. 3. Xwvs. concentration of polystyrene--cyclohexanesolutions (sample la, M,, = 163,000) for 35°, 45° and 65°. - from light scattering O from ultracentrifuge equilibria ® from critical miscibility measurements. Figure 3 shows Xwfor sample 1a (M w = 163,000) for the three temperatures. It reveals the very strong concentration dependence of X and its relatively small temperature dependence. This figure also shows a comparison with X determined by other methods: from ultracentrifuge equilibrium measurements on the same sample (O) <15) and from measurements of critical miscibility (®).~ts~ The values lie practically on the curves, the deviation being not larger than one unit in the third decimal place, which agrees with the experimental accuracy of our determination of x for this system. Results from osmotic pressure measurements published by Krigbaum and Geymer C19) and by Rehage and Palmen ~z°' 21) agree very well with these curves. Figure 4 shows the molecular weight dependence of Xw. There is only little difference between the x~, values for the different molecular weights. At 35 ° (the 0 temperature) and at 45 ° the difference is comparable with the accuracy of the determination of Xw. At 65 °, the difference between the X~ curves slightly exceeds the accuracy of Xw,

1070

TH. G. SCHOLTE 35 ° 45 °

0"60 65 ~ 0"58

0"56

× 0"54

0"52 5(1 la

0'50

7a

0 '48 I

I

0

0:1

w

i

1

0"2

0'3

FIG. 4. X,, of polystyrene--cyclohexane solutions for the different molecular weights of the polystyrene.

indicating t h a t this difference is real. The x , values exhibit s o m e w h a t greater differences for the different m o l e c u l a r weights. A t w = 0, (½ - - X) is p r o p o r t i o n a l to the second virial coefficient. It is well k n o w n t h a t the second virial coefficient decreases with increasing m o l e c u l a r weights (see for instance K r i g b a u m a n d F l o r y ' s results for p o l y i s o b u t y l e n e fractions in benzene (22~ a n d for polystyrene fractions in cyclohexane (23~ a n d the w o r k o f S o t o b a y a s h i a n d U e b e r r e i t e r (24) a n d o f Schulz, B a u m a n n a n d Darskus(25)). The initial values o f x at 65 ° for the different m o l e c u l a r weights agree with this dependence. T h o u g h this difference in X diminishes with increasing concentration, it is still discernible for c o n c e n t r a t i o n s up to 20 p e r cent. W i t h solutions o f the three polystyrene samples in toluene, the same c o n c e n t r a t i o n ranges c o u l d be covered in the light scattering measurements. Here we p e r f o r m e d the m e a s u r e m e n t s at 25 ° , 45 ° a n d 65 ° . Use was m a d e o f the refractive index increment values: (ts~ 25°: 45°: 65°:

dn/dw = 0"0952 + 0.0371 w dn/dw = 0.0992 - - 0.0374 w dn/dw = 0.1032 -I- 0"0379 w,

a n d the density values :(15 25°: O ----0-86178 + 0-1794 w + 0-0296 w 2 45°: O = 0-84311 + 0.1847 w q- 0-0303 w 2 65°: O = 0-82445 + 0.1903 w nL- 0.0307 w 2.

Determination of Thermodynamic Parameters of Polymer-Solvent Systems

1071

20--

O

-

x

-

7a

la

e~

tJ 0

0-I

FIG. 5.

O~/Owvs.

0-2

0.5

0

I

!

O" /

0.2

O

Oi

concentration of polystyrene-toluene solutions for the three polystyrene samples

O:

25°;

G:

45°;

A:

65 ° .

0.47

_

_..m.-

25*

0.45

0.4~

0.47

X

- -

0 " 4 5 --

0 . 4 3 --

0"47

7rI

- -

65*

~

5

a

0-45

0.43

-

I

I

I

0-1

0.2

0-5

w

FIG. 6. Xw of polysty~ne-toluene solutions for the different molecular weights of the polystyrene.

q~

0"00794 0.00783 0"00771

0.0159 0"0157 0-0155

0"0320 0"0316 0"0311

0"0482 0"0476 0"0469

0'0645 0"0637 0'0629

0"0810 0"0801 0-0790

0"1230 0"1216 0"1202

0"1660 0"1643 0"1625

0"2549 0"2527 0.2504

t (°C)

25 45 65

25 45 65

25 45 65

25 45 65

25 45 65

25 45 65

25 45 65

25 45 65

25 45 65

0"01

0"02

0"04

0"06

0"08

0"10

0"15

0"20

0"30

W

(g/g)

396 406"5 425

141"0 145"6 149"6

68"1 71"4 74"6

26"95 28"57 29"92

16"90" 18"07 18"96

9"58 10-32 10"84

4"53 4"96 5"24

1"48 1"64 1-77

0"60 0-64 0"70

7a ( M ~ =: 51000)

126"6 132"3 139-6

59"9 63"1 67"2

22"12 23"46 24"60

13"19 13"95 14"52

6"96 7"37 7'68

2"98 3-14 3"34

0-79 0"86 0-94

0"255 0"28 0"32

la (M. = 163000)

20"97 22"66 23-20

12"20 13-18 13"45

6"27 6"68 6"90

2"53 2"66 2-79

0-58 0"61 0-67

0-16 0"17 0"18

5a ( M . =520000)

--A/~ o x 10 -6 (crg/mole)

0"4584 O'4649 0-4676

I)-4457 0"4502 0"4548

0-4467 0"4487 0"4507

0"4464 0"4469 0-4484

0"4455 0'4451 0"4466

0"4448 0"4436 0"4450

0"4464 0"4431 0'4441

0-4511 0'4451 0-4427

0"4476 0-4476 0"4405

7a (M~ = 51000)

I)'4541) 0"4564 0"4575

0"4530 0"4544 0'4541

0"4530 0-4533 0"4546

0"4529 0"4536 0-4552

0"4529 0-4534 0"4548

0"4535 0"4544 0"4544

0-4571 0"4554 0"4530

0"4604 0'4572 0"4494

la (M. = 163000)

X,v

0"4539 0"4527 0-4559

0-4545 0"4534 0-4563

0"4548 0"4548 0-4568

0-4559 0-4568 0"4577

0-4603 0"4612 0"4592

0-4627 0.4629 0"4632

5a ( M ~ -520000)

0"3682 0-3724 0.3708

0'3702 0"3722 0"3740

0-3824 0"3806 0"3796

0"3918 0"3891 0"3875

0"3949 0"3909 0-3892

0-3984 0"3934 0"3917

0-4051 0-3972 0.3949

0"4166 0-4046 0-3974

0-4130 0"4103 0"3955

7a ( M . := 51000)

TABLE 2. CItEMICAL POTENTIALS AND INTERACTION PARAMF.TI!I;tS OF POLYSTYRENE-TOLUENE SOLUTIONS

0"3823 0-3816 0"3781

0.3915 0"3895 0"3848

0-4021 0-3994 0"3975

0"4064 0"4043 0"4035

0"4112 0"4092 0-4083

0"4168 0.4156 0"4125

0'4268 0'4219 0'4154

0'4344 0"4275 0"4117

la ( M . -163000)

X4~

0-4035 0"3984 0-3995

0-4042 0"4056

0'4089

0"4112

0"4139 0-4112

0"4203 0"4192 0"4177

0"4316 0"4308 0"4252

0"4374 0"4359 0"4342

5a ( M ~ ;-. 520000)

m

0

Oq

P

,-4

1-9

Determination of Thermodynamic Parameters of Polymer-Solvent Systems

1073

Figure 5 shows 81x/Sw vs. w for the three samples. The system p o l y s t y r e n e - t o l u e n e shows behaviour quite different from that of polystyrene-cyclohexane. Since toluene is a good solvent, 8 (~/x)/Sw begins to increase at lower concentrations a n d the values are, on the whole, much greater. The temperature dependence of 8/~/8w is very small. Table 2 gives ~ o and ~ a n d also ~ a n d X, for the various samples a n d temperatures. Figure 6 shows X~ as a f u n c t i o n of concentration. As is already known, its' .,6~ the c o n c e n t r a t i o n dependence of x is very small for this system. The molecular weight dependence of x is greater than for the p o l y s t y r e n e cyclohexane mixtures, b u t it tends to decrease toward higher concentrations. Acknowledgements--The author is indebted to Mrs. E. M. L. Nijsten-Colaris and Mr. H. M. Schoffeleers for the light scattering measurements and to Mr. N. L. J. Meijerink for the measurements of refractive index increment and density values.

REFERENCES (1) G. Oster, Chem. Rev. 43, 319 (1948). (2) G. Oster, In: Technique of Organic Chemistry, Vol. 1, Part II, (edited by A. Weissberger), New York (1960). (3) K. A. Stacey, Light Scatterin~ in Physical Chemistry, Butterworths Scientific Publications, London (1956). (4) A. Einstein, Ann. Phys. 33, 1275 (1910). (5) M. Smoluchowski, Ann. Phys. 25, 205 (1908). (6) B. H. Zimm and P. M. Doty, J. chem. Phys. 12, 203 (1944). (7) H. C. Brinkman and J. J. Hermans, J. chem. Phys. 17, 574 (1949). (8) F. Zernike, Thesis Amsterdam (1915); Arch. Neerland. (3A) 4, 74 (1918). (9) J. G. Kirkwood and R. J. Goldberg, J. chem. Phys. 18, 54 (1950). (10) W. H. Stockmayer, J. chem. Phys. 18, 58 (1950). (11) P. Debye, J. chem. Phys. 31, 680 (1959). (12) P. J. Flory, J. chem. Phys. 10, 51 (1942). (13) M. L. Huggins, Ann. N. Y. Acad. Sci. 43, 1 (1942). (14) P. J. Flory, Principles of Polymer Chemistry, Cornell University Press (1953). (15) Th. G. Scholte, J. Polym. Sci. A2, (8) 1970. (16) R. Koningsveld and A. J. Staverman, J. Polym. Sci. A2, 6, 325 (1968). (17) M. Gordon, H. A. G. Chermin and R. Koningsveld, Macromolecules 2, 207 (1969). (18) R. Koningsveld, L. A. Kleintjens and A. R. Shultz, to be published. (19) W. R. Krigbaum and D. O. Geymer, J. Am. chem. Soc. 81, 1859 (1959). (20) H. J. Palmen, Thesis Aachen (1965). (21) G. Rehage, H. J. Palmen, D. M/511erand W. Wefers, to be published. (22) W. R. Krigbaum and P. J. Flory, J. Am. chem. Soc. 75, 5254 (1953). (23) W. R. Krigbaum, J. Am. chem. Soc. 76, 3758 (1954). (24) H. Sotobayashi and K. Ueberreiter, J. Polym. Sci. A2, 1257 (1964). (25) G. V. Schulz, H. Baumann and R. Darskus, J. phys. Chem. 70, 3647 (1966). (26) G. Rehage, KolloidZ. 196, 97 (1964). R~sum~---Apartir des intensitrs de la lumi~re diffusre par des solutions dilures ou concentrres d'un polymrre dormr, les potentiels chimiques du polymrre et du solvant peuvent f:tre d~terminrs. Pour diffrrentes temprratures et pour des concentrations altant jusqu'~t 30 pour cent en poids, des experiences ont ~t~ effectu~es avec des solutions de polystyr~nes monodisperses de diffrrentes masses molrculaires, soit dans un mauvais solvant (cyclohexane) soit dans un bon (toluene). Pour toutes les concentrations, le param~tre d'interaction x drpendait faiblement de la masse molrculaire. Les rrsultats sont comparrs ~ ceux d&erminrs par des mesures d'ultracentrifugation ~. l'rquilibre et de miscibilit~ critique. Sommario--Dall'intensitb. della luce di scattering da soluzioni diluite e concentrate di un dato polimero, pub essere determinato il potenziale chimico del polimero e del solvente. Sono stati fatti esperimenti con soluzioni di polistireni monodispersi di diversi pesi molecolari in solventi poveri (cicloesano) e in solventi buoni (toluene) a differenti temperature e concentrazioni che

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arrivano al 30 ~ in peso. Per tutte le concentrazioni, il parametro x di interazione non ha una grande dipendenza dal peso molecolare. I risultati sono raffrontati con quelli determinati da misure di ultracentrifugazione di equilibrio e di miscibilittt critica. Zttc~mmenfassnng--Aus der Intensittit des von verdtinnten und konzentrierten LSsungen eines gegebenen Polymeren gestreuten Lichts, l~.lBtsich alas chemische Potential des Polymeren und des L6sungsmittels bestimmen. Mit LtSsungen von monodispersen Polystyrolen mit verschiedenen Molekulargewichten wurden Versuche in einem schlechten L6sungsmittel (Cyclohexan) und in einem guten L/Ssungsmittel (Toluol) bei verschiedenen Temperaturen und bei Ko~entrationen his du 30 Gewichtsprozent durchgef'tihrt. Bei allen Konzentrationen scheint der Wechselwirkungsparameter geringftigig abhttngig zu sein vom Molekulargewicht. Die Ergebnisse werden verglichen mit denen, die durch GleichgewiehtsUltrazentrifugation und Messungen der kritischen Mischbarkeit bestimmt wurden.