A system-independent correlation between the enthalpy and the entropy of dilution for polymer solutions

European Polymer Journal, 1973, Vol. 9, pp. 723-732. Pergamon Press. Printed in England.

A SYSTEM-INDEPENDENT CORRELATION BETWEEN "IHE ENTHALPY AND THE ENTROPY OF DILUTION FOR POLYMER SOLUTIONS M. D. LECHNER* and G. V. SCHtJLZ Institut ffir physikalische Chemie der Universit/it Mainz (W. Germany), Sonderforschungsbereich "Makromolekfile", Mainz/Darmstadt, Germany (Received 28 December 1972)

Abstract--The second osmotic virial coefficient (A2) and its entropic and enthalpic parts (Az.s and Az.u) have been determined, by means of light-scattering measurements, for solutions of polystyrene, polymethylmethacrylate and cellulose nitrate of different molecular weights in 19 solvents. A distinct qualitative correlation exists between A2 and Az.n and between A2.s and A2.n. The elimination of the "geometric" parameters of the polymer, by dividing these coefficientsby suitably chosen reduction parameters, shows that the reduced coefficients obtained .42° and .42.s° are predominantly functions of the reduced enthalpy coefficient A2.n°. 1. I N T R O D U C T I O N A SERmS of experimental results indicates that relations exist between the enthalpic and the entrophic contribution of the second osmotic virial coefficient A2, defined by the equation, A2 = A2,~ + A2,s. (1) which are to a certain extent independent of the particular system. (1-7) A wide range of experimental data has been obtained for polystyrene, polymethylmethacrylate and cellulose nitrate in 19 solvents including both highly exothermal and highly endothermal systems. The results are examined for correlations between the coefficients of Eqn. (1). The excess entropy of dilution of a polymer solution depends on two groups o f factors, (a) the size, form and flexibility of the macromolecules, (b) the energetic interactions between like and unlike molecules or segments in the solution, which are, for instance, represented by the parameter w ----2 E12-~1~-E22 t 8,9) in the case of the lattice theory, or by a minimum or a step in the potential curve, in the case of the excluded volume theory. For athermal solvents, w = 0 and the potential function shows neither a pronounced minimum nor a step. In this case, the second osmotic virial coefficient is exclusively determined by the "geometric" factors listed under (a). It is therefore possible to calculate A 2 ath : A2, s for athermal solutions to a good approximation from the parameters of the polymer.(3) For positive or negative heats of dilution, A2,s is increased or decreased over the calculated value in a characteristic manner. Subsequently a semiquantitative but very general relation between A2,s and AE,H is observed. The second osmotic virial coefficient (A2) is given by the equation for the osmotic pressure (rr) rr c

RT M + A2c + . . . .

(2)

* Present address: Fachhochschule Aalen, 708 Aalen (W. Germany), Fachbereich Chemic. Abt. Physikalische Chemie. 723

724

M . D . LECHNER and G. V. SCHULZ

where M is the molecular weight of the polymer, c the concentration in g/cm a and R T has its usual meaning. The chemical potential of the solvent changes on the transfer from the pure phase to the solution by a/z~ = --rrg~. Introducing the excess potential At~I ~ = At~I -- R T In xl

(3)

(x~ = mole fraction of the solvent) and the excess entropy AS1 s = AS1 + R In xl

(4)

so that A/~IE ---- AH1 -- T AS1 E, one obtains for sufficiently dilute solutions A2 = --At~IE/RT ~1c 2

(5a)

- - A H 1 / R T ~1c 2

(5b)

Az,s = ÷ A S I E / R ~1c 2.

(5c)

A 2 . 1 t ~--

(AHI = partial molar heat of dilution; vl = partial molar volume of the solvent). The following considerations refer to dilute solutions for which Eqns. (5) are applicable. The experimental results were obtained by measurements of the intensity of scattered light at atmospheric pressure. The data refer to a constant temperature of 25 °. The experimental values of A2, A2,s and A2,~/are compiled and compared in 2; a qualitative correlation is already apparent. By referring these values to the athermal solution, the reduced quantities A2°, A2,H° and A2,s ° are introduced in 3. A stronger correlation exists between these quantities which is to a great extent independent of the particular system. 2. INTERRELATION BETWEEN A2, A2,rl AND A2,s IN DILUTE POLYMER SOLUTIONS The measurement of the temperature dependence of A2 allows determinations of the enthalpy coefficient according to the Eqn. ( l y 1).

in which all quantities on the right-hand side are accessible (a~ = thermal expansion coefficient). The values for solutions of polystyrene, polymethylmethacrylate and cellulose nitrate samples of different nitrogen contents (CTN 13.9 and CTN 12.9) were obtained in this manner. Experimental details have been given in the cited papers. In Fig. 1, A2 is given as a function of A2.n, in Fig. 2, A2,s as a function of A2.u. A clear trend is recognizable: all pairs of values lie within a relatively narrow band, in spite of the fact that we are dealing with very dissimilar systems. Pure hydrocarbons and strongly polar solvents are included and the same is true for the polymers. The range runs from highly endothermal to highly exothermal solutions and from positive values of the excess entropy to considerably negative ones. The diagonal in Fig. 2 separates the solutions with positive from those with negative

A System-independentCorrelationfor PolymerSolutions

725

I

1

I

V t. . . . .

!

II

!

1

x

E i

i

I

i

T

L

I

I

I !

-i-20 -18I

-16

-14

-12

-10

-8

-6

-4

-2

0

J

2

4

6

I

8

I 10

12

14

16

i

18

i i

20

,

22

24

28

FIG. 1. The second osmotic virial coefficient (A,) measured by light scattering, as a function of the enthalpy coefficient (A2.u) for 50 systems at 2Y. Polystyrene [3, 13, 14, 15, 18] [] ff/lw = 0.26 x 10s; [] ff'lw = 0.58 × 10s; F?] IFlw -: 1.7 < 105; [] l~Iw = 1.8 × 10s; [] IV-I,, = 2.1 x 105; [] IVlw = 5.7 x 10s; [] ff,lw = 1-6 x 106; !] 1Vlw = 3.9 x 106; [] l~lw = 6.0 x 106; p o l y m e t h y l m e t h a c r y l a t e [3, 16, 18] 4) ff-lw -0.3 x 10s; © IFIw = 2"I x 10s; cellulose trinitrate (13"9%N) [4, 17] , i IFlw 0-82 x 10s; .~ ff,lw = 1"6 x 10s; A IF,I,, = 3.5 x 10s; ix, ~ w = 7 . 4 x 10s; /~ ff'lw = 1"66 x 106: cellulose trinitrate (12.9%N) [17] ~¢ 191w = 1"4 x 10s; V 1~I~ = 2"9 x 10s; 7 IFIW ~ 6.2 x 10s; lie ff'lw = 1.70 x 106.

Solvents: (1)cyclohexane; (2) dekalin; (3) isoamyl acetate; (4) butyl chloride; (5)diethyl oxalate; (6) hexyl-m-xylol; (7) n-butyl formate; (8) 4-heptanone; (9) malonic ester; (10) butyl acetate; (11) methyl isovalerate; (12) 2-methyl-4-pentanone; (13) ethyl acetate: (14) t e t r a h y d r o f u r a n ; (15) toluene; (16) c h l o r o f o r m ; (17) isopropyl proprionate, (18) methylethylketone; (19) acetone. A 2 values. This line can only be undercut by small amounts in the case of high polymer

solutions since phase separation sets in. The correlation which is indicated above this line can be worked out somewhat more distinctly, as will be shown in the next section.

3. INTERRELATION BETWEEN THE REDUCED QUANTITIES A2°, 32,/_/0 AND A2.s° The polymers dealt with in Section 2 differ widely in their geometrical form, in their flexibility and in the size of the molecules. One can expect to obtain a correlation between the coefficients A2, A2,n and A2,s independent of the individual character of the system, by eliminating the individual properties of the particular polymer. This can be done in the following way: the measured A2, A2,H and A2,s values are reduced in two respects: (1) in order to eliminate the difference in molecular weight for a given type of polymer, we divide the measured A2, A2,H and A2.s values by the h(z) function valid for the particular system.

726

M. D. LECHNER and G. V. SCHULZ

3

FIG. 2. Relation between the entropy and the enthalpy coefficientfor systems as in Fig. 1. (2) In order to be in a position to compare different types of polymers, we additionally divide the experimental values by the geometrical factor (~sp/Mo)(rra[4d) deduced for athermal solutions. ~3,a) From this we deduce

A2* = (~JMo)(Tra/4d)h(z),

(7)

and for the reduced quantities

~20 = ~2/z42"

[~a)

A2m ° : A2m/A f f

(8b)

~2,s/~2"°

(8c)

~2,s 0 :

t~,p is the partial specific volume of the polymer, Mo and a are the molecular weight and the X-ray length of the monomer unit, respectively, and d is the mean diameter of the polymer chain, which can be evaluated from the three molecular parameters mentioned above, by means of the equation. Mo~sp = NA(rr/4)ad 2.

(9)

The right-hand side of Eqn. (7) therefore consists of data which can easily be obtained, except the h(z) function, for the polymer. The introduction of the h(z) function in Eqn. (7) takes into account the influence of

A System-independent Correlation for Polymer Solutions

727

the molecular weight and of the stiffness of the polymer; the reduced values (A 2°, etc.) therefore refer to short chains for which z ~ 0 and consequently h(z) ~ 1. The h(z) function can be obtained from the experimental data in the following manner. According to the two parameter theory °°)

A2 = (Na/2)(fi/Mo2)h(z)

(lO)

z ---- (47r)- a/2(fl/Mo2)M I/2K o- a/2.

(ll)

Ko = /M

(12)

with

For each polymer

is a characteristic constant taking into account its flexibility (or stiffness); < re2> is the mean square radius of gyration at the theta temperature (for which A2 = 0). From Eqns. (10)-(12) one obtains

zh(z) = A2M'/2Ko - a/2(4rra/2N a)- '.

(13)

The quantities on the right-hand side of Eqn. (13) are accessible from light scattering measurements. To evaluate h(z) and zh(z), one needs a generally applicable h(z) function. According to the present experience(a,4,11) the function of Casassa (12) is the most universal:

h(z) --=

1 -- exp[-- 5.68z/a2 a] 5"68z/a23

(14a)

z = (a22 -- 1) a23]2"043.

(14b)

Taking a 2 as a running variable, one calculates a series of pairs of values of the functions h(z) and zh(z) and can then transfer the zh(z) values calculated according to Eqn. (13) into the corresponding h(z) values by means of an appropriate graph (Fig. 3). 1'4 1"2 h{z) 1.0 0-8

\

\

\

0,6 04,

\\

0.2 0 -0,2

0

0"2

0"4

0"5

0"8

1'0

1"2 1"L ,,. Z h(z)

1-5

FIG. 3. h(z) as a function of zh(z) according to the relation of Casassa; Eqns. (14a) and (14b). A comparison o f Eqns. (7) and (13) shows that A2* is exclusively determined by the properties of the polymer, namely vsv, a/Mo, I(o and M, whereby one has to use a special function for h(z). Since t~sv and a/Mo can be determined very precisely, and

-13"l

-(

1'4

1-8

I

-1'0

I

-08

I

~'A~, H

-06 -04

-0'2

0

0"2

04

06

0"8

l'O

1"2

1"4

1:6

1"8

2-0

2-2

2.4

2"6

2"8

3~1

3"2

3"4

36

3"8

4'0

4'2

/dl

4"6

FIG. 4. R e d u c e d virial coefficient (A2 °) as a f u n c t i o n o f t h e reduced e n t h a l p y coefficient (A2,u°); for s y m b o l s , see Fig. 1. P r e s s u r e : 1 a t m ; temp. 25 ° .

-1'2

I

I

I I I

t-fll~

4-8

N

O~

.<

Z m

r~

t7

A System-independentCorrelation for Polymer Solutions

729

since the error in M and A2 as determined from light scattering measurements is small, the certainty in the evaluation of A2* mainly depends on the reliability of the h(z) function used and on the accuracy in the determination of OA2/t3T and Ko. Taking these considerations into account, the reduced values in the Eqns. (8a)--(8c) are subject to an error o f approximately 4-0.2. TABLE 1. PARAMETERS OF POLYMERS

Polymer Polystyrene Polymethylmethacrylate Cellulose nitrate (CTN 13"9) Cellulose nitrate (CTN 12.9)

K o × 1018 (cm2molg- 1)

~p

a × 108

Mo

(cm3/g)

(cm)

(g/mol)

0.92 0.81 0"57

2.54 2.54 5"13

104 100 294

7.75 4.8 99

0.57

5'13

285

53

With the aid of the Eqns. (7), (13)and (5a)-(5c)and of the numerical values given in Table 1 for the polymers under discussion, the reduced coefficients A2°, A2,H° and A2,s ° have been evaluated. They are plotted in Figs. 4 and 5. One can see from Figs. 4 and 5 that the reduced coefficients show a considerably better correlation than the original values in Figs. 1 and 2. The deviations from a common line scarcely lie outside the limits given Lby the measuring errors and the uncertainty in the h(z) function.

FIG. 5. Reduced entropy coefficient (A2,s°) as a function of the reduced enthalpy coefficient (A~,n°).

730

M. D. L E C H N E R and G. V. SCHULZ TABLE 2. LIST OF THE EXPERIMENTALVALUESOF THE SYSTEMSUSEDIN FIGS. 1, 2, 4, 5

A2 Solvent

i~lw x 10 -6

A2.1t

A2.s

(cm a mol g-2) 10-4

Polystyrene 0.17 --1.52 --70.0 0.17 --0.28 --20.0 0.17 --1.00 --12.55 0.026 0.41 --20.5 0.058 0'30 --17.6 0.18 0.44 --12.5 0.57 0'41 --11.9 1.60 0.41 --12.2 6.00 0.35 --11.9 n-Butyl formate 0'17 1"43 --9"4 Malonester 0'026 --0"29 --9"54 Malonic ester 0.058 --0.26 --8.94 0.18 --0.24 --8"94 0.57 --0.23 --8.33 1.60 --0.13 --7.45 6.00 --0.11 --7.15 Toluene 0.21 4.2 0 0-57 3.3 0.3 1.60 2.46 0.2 3-92 1.87 0.1 Tetrahydrofuran 0.026 8"35 5"52 0"053 6"65 4"92 0"165 4"96 3"66 0'54 3"79 2.94 1"50 2"88 2"49 5'70 2'11 1"34 Chloroform 0"18 4"5 2.82 Polymethyl methacrylate Isoamyl acetate 0"21 --0"90 --13"7 1-Chlorbutane 0'03 --0.32 --14"6 0"21 --0"40 --12"8 4-Heptanone 0"21 --0.50 --9"3 Butyl acetate 0"21 0"70 --4"5 Methyl isovalerate 0"21 0"80 --4"0 2-Methyl-4pentanone 0"21 1"10 --3"6 Ethyl acetate 0"21 1.71 --0"95 Tetrahydrofuran 0"20 3"25 --0.1 Chloroform 0.21 4'53 2'8 Cyclohexane Dekalin Diethyloxalate Hexyl-m-Xylol

A2 0

A2,H 0

h2,s 0

Refs.

68-5 19.7 11.55 20.9 17.9 12.9 12-3 12.6 12.3 10"8 9.25 8.70 8.70 8.10 7.32 7.04 4.2 3.0 2.26 1.77 2'83 1"73 1'30 0'85 0.39 0.77 1"70

--0.0362 --0.0125 --0.0323 0.022 0-016 0.026 0"027 0.032 0.036 0"119 --0'014 --0.012 --0.011 --0.010 --0.005 --0.004 0.875 0.93 0.88 0.82 1.17 1-10 1.07 1'12 1'05 1"06 0"854

--1.666 --0.893 --0.405 --0.110 --0.95 --0.75 --0.79 --0.96 --1.23 --0.784 --0.460 --0.426 --0.426 --0.350 --0.315 --0.237 0 0.09 0-07 0'04 0.395 0.287 0"280 0"250 0"214 0'385 0"533

1.630 0.880 0.373 1.12 0.96 0.78 0.82 0.99 1.27 0.900 0"45 0.414 0.415 0.340 0.310 0.269 0.875 0.84 0.81 0.78 0.770 0"816 0"790 0"865 0.909 0"670 0"321

13 13 13 15 15 15 15 15 15 15 15 15 15 15 15 15 3.15 3.15 3.15 3.15 14 14 14 14 14 14 18

12'8 14'3 12"4 8"8 5"2 4"8

--0"023 --0"015 --0"014 --0.015 0"063 0"078

--0"357 --0"677 --0.460 --0"283 --0"406 --0"392

0.334 0"662 0"450 0.268 0"468 0"470

16 16 16 16 16 16

4"7 2"66 3-35 1"73

0'133 0"31 1.06 1"97

--0"436 --0"17 --0"03 1"22

0"570 0"48 1"08 0"75

16 18 3 18

3"4 2"35 0"15 1"1 --1.1 +1"4 -t-0-7

1"41 1"30 1"22 1"26 1.20 1.24 1-02 '

1"00 1"02 1.20 1.05 1.33 1"07 0"93

0'41 0.28 0.02 0.19 0.13 0.17 0"09

4-0 1"4 --1"5 --4"4 --9"1 --7"3 --13-5

1"66 1"68 1"68 1"78 1"58 1"79 2-71

1"66 1"48 1"89 2"36 2"86 2"96 5"65

4-0 0-20 --0"21 --0"58 --1"28 --1"17 --2"94

Cellulose nitrate (13"9 Yo N) Isopropyl proprionate Methylethylketone Ethyl acetate Acetone Acetone Acetone Acetone

0.35 0.35 0.35 0.082 0-16 0.74 1"66

11"8 8"4 11-0 8"65 10"3 10"15 10"3 11"4 10-3 11.4 10"0 8"6 8"4 7"7 Cellulose nitrate (12.9 Yo N)

Isopropyl proprionate 0.29 Methylethylketone 0.29 Ethyl acetate 0.29 Acetone 0.14 Acetone 0.29 Acetone 0.62 Acetone 1.70

11"8 11 "9 11"9 13"3 11"2 11"2 12"5

11"8 10"5 13"4 17.7 20"3 19"5 26-0

4 4 4 4"17 4'17 4.17 4"17

17 17 17 17 17 17 17

A System-independent Correlation for Polymer Solutions

731

The introduction o f the h(z) function in Eqn. (7) means, as already mentioned, that the influence o f the molecular weight is eliminated or more precisely that the excess values manifesting themselves in A2, A2.n and A2,s are reduced to such short chains that z vanishes and consequently h(z) approaches unity. I n particular, one can see the following facts f r o m the Figs. 4 and 5. F o r A2,n ° ---- O, the values o f A2 ° and A2.s° lie, as one would expect, close to 1. For negative A2,n ° values (endothermal solution), A2 ° rapidly falls to values near zero. This is a consequence o f the fact that the negative A2,H° values are approximately compensated by A2,s ° values o f nearly the same magnitude in this region. In the exothermal range (A2,H° > 0) only a slight increase in A2 ° can be observed and A2,s ° decreases with a slope somewhat less than --1 to highly negative values. This trend does not depend on the special system. In the exothermal region A2 ° ~ 1 + 0.20 Az,n ° and Az,s ° ~ 1 -- 0"8 A2,H 0

with a scattering which hardly exceeds :k0" 15. In the endothermal region, A2 ° decreases from the value 1 to nearly 0 within approx. 0.4 units and A2,s ° balances out to A2,s ° ~ A2,u ° in this range. Generally, one can say that, in all systems treated in this paper, the coefficients A2 and A2.s are practically fixed by the "geometric" parameters o f the polymer appearing in Eqn. (7) and by the heat o f dilution as manifested in A2.n.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)

REFERENCES G. V. Schulz, H. Inagaki and R. Kirste, Z. phys. Chem. NF24, 390 (1960). G. V. Schulz, A. Haug and R. Kirste, Z. phys. Chem. NF 38, 1 (1963). G. V. Schulz, H. Baumann and R. Darskus, J. phys. Chem. 70, 3647 (1966). E. Penzel and G. V. Schulz, Makromolek. Chem. 162, 269 (1972). M. Lechner, Ber. Bunsenges. 75, 50 (1971). B. A. Wolf, Ber. Bunsenges. 75, 924 (1971). B. A. Wolf, J. Polym. Sci. A2 10, 847 (1972). R. H. Fowler and G. S. Rushbrooke, Trans. Faraday Soc. 34, 1272 (1937). E. A. Guggenheim, Mixtures. Oxford (1952). B. H. Zimm, W. H. Stockmayer and M. Fixman, J. chem. Phys. 21, 1716 (1953). C. G. Berry, J. chem. Phys. 44, 4550 (1966). E. F. Casassa, J. chem. Phys. A2, 31, 800 (1959). G. V. Schulz and H. Baumann, Makromolek. Chem. 60, 120 (1963). G. V. Schulz and H. Baumann, Makromolek. Chem. 114, 122 (1968). H. Baumann, Dissertation, Mainz (1965). G. V. Schulz and R. Kirste, Z. phys. Chem. NF27, 301 (1969); 30, 171 (1969). G. V. Schulz and E. Penzel, Makromolek. Chem. 112, 260 (1968). M. D. Lechner, and G. V. Schulz Makromolek. Chem., in preparation.

R6sum6----On a d6termin6 le second coefficient du viriel osmotique (,42) ainsi que ses composantes entropique et enthalpique (,42,s et A2.r,) par diffusion de la lumi~re, pour des solutions, darts 19 solvants, de polystyrene de poly(m6thacrylate de m6thyle) et de nitrate de cellulose de diff6rents poids moMculaires. I1 existe une corr61ation qualitative distincte entre A2 et A2.n et entre A2,set ,42.n. On 61imine les param~tres "g6om6triques" du polym~re, en divisant ces coefficients par des param~tres de r6duction convenablement choisis. On montre ainsi que les coefficients r6duits ainsi obtenus (A2° et A2°s) sont essentiellement fonction du coefficient d'enthalpie r6duite (A2°,n).

732

M . D . L E C H N E R and G. V. SCHULZ

Sommario---Mediante misurazioni di dispersione luminosa, si sono determinati il secondo coefficiente viriale osmotico (A2) e le sue parti entropica ed entalpica (A2.s e A2,n) per soluzioni di polistirene, polimetilmetacrilato e nitrato di cellulosa, con differenti pesi molecolari, in 19 solventi. Tra Az e A2.n e tra A2,s e ,'tz.n esiste una distinta correlazione. L'eliminazione dei parametri "geometrici" del polimero, dividendo questi coefficienti con parametri di riduzione idonei, mostra che i coefficientj ridotti ottenuti (A2° e A2,s °) sono in prevalenza in funzione del coefficiente di entalpia ridotto (A2.n°). Zusanunenfassung--Der zweite osmotische Virialkoeffizient A2 und sein enthalpischer und entropischer Anteil (A2.s und A2.~) wurden durch Lichtstreuungsmessungen an L/Ssungen von Polystyrol, Polymethylmethacrylat und Cellulosenitraten verschiedenen Molekulargewichts in insgesamt 19 L6sungsmitteln bestimmt, Es existiert eine deutliche Korrelation zwischen A2 und A,,n sowie zwischen A2.s und A2.n. Die Eliminierung geometrischer Unterschiede der Polymeren, indem man die Koeffizienten durch geeignete Parameter dividiert, zeigt, dass die so reduzierten Koeffizienten A2 ° und A2.s ° iiberwiegend Funktionen des reduzierten Enthalpiekoeffizienten A2.H° sind,