Lattice-Hole Theory: Bulk Properties and Surface Tension of Oligomers and Polymers

Journal of Colloid and Interface Science 216, 424 – 428 (1999) Article ID jcis.1999.6316, available online at http://www.idealibrary.com on

NOTE Lattice-Hole Theory: Bulk Properties and Surface Tension of Oligomers and Polymers A scheme for the prediction of surface tension from bulk properties of polymer melts was previously developed. It is based on a combination of Guggenheim’s empirical temperature function with a hole or free volume fraction, derived in lattice-hole theory. This approach is now extended to oligomeric species, typified by n-alkanes. Successful results for surface tension and its temperature coefficient ensue. As to further applications to high polymers, a satisfactory outcome is obtained for absolute values and temperature coefficients in the majority of the numerous instances of structurally diverse systems. This encourages predictions for some high T g systems, and mixtures, where experimental information is not available. © 1999 Academic Press Key Words: surface tension; lattice-hole theory; free volume; oligomers; polymers.

INTRODUCTION AND RECAPITULATION The relation between the bulk thermodynamic properties of both low and high molar mass melts and their surface tension has been considered by several authors. There are two types of approaches. In one the connection arises naturally from a theoretical formulation which involves a theory of the bulk properties (1– 6). The other approach deals with correlations and scaling properties (7, 8). Carri and Simha (CS) examined the relation between surface tension and bulk thermodynamics of polymers in terms of lattice-hole theory of the latter (9). A central quantity here is the hole or free volume fraction, h(T, P), determined by the minimization of a configurational free energy. Its relation to the surface tension g (T) is obtained by elimination of T between h and g. A linear relation ensues, viz.,

g ~T! 5 A 1 Bh~T!

[1]

with A and B system characteristic parameters. Deviations from Eq. [1] become noticeable at elevated temperatures or h-values exceeding about 0.25. Proceeding further, CS considered the empirical representation of g (T) by Guggenheim’s two parameter expression (10):

S

g ~T! 5 g 0 1 2

T Tc

D

11/9

.

[2]

The scaling temperature T c for low molar mass systems has been identified with the critical temperature. In the case of polymers studied by Dee and Sauer (11, 12), CS fitted Eq. [2] as a two parameter expression with reasonable success (9). Deviations from Eq. [2] have been noted by Utracki which affect the evaluation of temperature coefficients (13). See also below. Based on lattice-hole equation of state (EOS) (14), expressions for g 0 and T c have been developed (9). The postulated coexistence between Eqs. [2] and [1] with h(T) 0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

given by theory and the experimental EOS results in a remarkable invariance of the scaled quantities A/ g 0 and B/ g 0 . The purposes of this note are as follows: First, to explore chain-molecular systems of low molar mass, typically the homologous series of n-alkanes, in the frame of the approach described, and second, to examine the pertinence of the relations obtained in Ref. (9) for further polymer systems and to predict the surface tension for others.

NORMAL ALKANES We begin with a consideration of Guggenheim’s Eq. [2]. In the present application and to simplify matters we adopt the values of the critical temperatures for T c . The chainlengths n vary from 6 to 20 with a temperature range of approximately 100 degrees. From the g (T) published in the literature (15) there follow the g 0 values. A typical result is illustrated by Table 1 for n-nonane. A slight, although systematic, decrease of the “constant” g 0 with increasing temperature (2% over 100 degrees) is noticeable. An average of 52.4 mN/m generates a maximum deviation of the recomputed g equal to 2% at the highest temperature. Proceeding in the identical manner for the other members of the series, one can define an overall average ^g 0& 5 51.9 mN/m with a standard deviation of 0.6. Recomputing g once more, we observe a maximum deviation from the experiment equal to 64% for n 5 6 to 20. In light of these results we continue to use Eq. [2] as an adequate representation of the absolute values of g, although not of temperature coefficients. Next, consider the relation between g (T) and h(T), derived from the EOS. For the present purposes we recall the two alternative procedures employed in the evaluation of the theoretical scaled versus the experimental EOS, depending on the definition of the chain segment and on the manipulation of the “flexibility” parameter c/s (9), where 3c is the number of external degrees of freedom in an s-mer. One treatment (I) assigns an a priori value, for example, 3c 5 s 1 3, and ultimately obtains the relation between the “effective” segment of the s-mer and the chemical repeat unit of the chain. The other treatment (II) identifies the segment with the repeat unit, distinguishes between terminal and internal units for sufficiently short chains, and deals with c as a disposable parameter. Both methods have been employed for the n-alkane series, the former in Refs. (16 –18) and the latter in Refs. (19, 20). The latter is particularly valuable in establishing group contributions. Both treatments are applied below. This serves two purposes: One is to examine the effect of numerical procedures in the EOS analysis on the prediction of g. The other is to allow comparisons with the results obtained for polymers (9), where primarily method I has been employed. To evaluate Eq. [1], the relations between temperature T, scaled temperature T˜ 5 T/T*, and h are required. The scaling temperature T* and, for future reference, the scaling pressure P* are displayed in Table 2. These quantities are derived from the EOS coupled with the minimization condition (14). Based on a fixed ratio 3c/(s 1 3) 5 1, we obtain s 5 0.394 1 0.352n (18), with n 5 6 to 17. The h-values so obtained can be fitted to a quadratic form in T˜ , viz.,

424

h~T˜ ! 5 a~n!T˜ 2 1 b~n!T˜ 1 c~n!; 6 # n # 17,

[3-I]

425

NOTE

TABLE 1 Surface Tension g and g 0, Eq. [2] for n-Nonane t(°C)

g exp (mN/m)

g 0 (mN/m)

g a (mN/m)

24.8 23.9 23.0 22.5 22.0 21.1 20.1 17.8 15.4

52.5 52.6 52.7 52.6 52.5 52.6 52.3 52.1 51.4

24.7 23.8 22.9 22.4 22.0 21.0 20.1 17.9 15.7

0 10 20 25 30 40 50 75 100 Mean ^g 0& 52.4 a

Computed from Eq. [2] with g 0 5 52.4.

with 2.8 # T˜ 3 10 2 # 7.5. This covers the actual T range encountered in surface tension data. The corresponding relation derived from Yahsi’s computations (19) with adjustable c is h~T˜ ! 5 b 0 ~n!T˜ 1 c 0 ~n!; 12 # n # 20

[3-II]

with the linear fit over the more limited T˜ range, 2.2 # T˜ 3 10 2 # 3.6. The coefficients of Eqs. [3-I, 3-II] are displayed in Table 2. Linearity of Eq. [1] is preserved by both methods. In Table 3 the critical temperature, T c , the coefficients A and B, and the characteristic ratios A/ g 0 and 2B/ g 0 are listed where g 0 is the temperature averaged value; see comments following Eq. [2]. A systematic trend with n is noticeable. As a matter of comparison we recall for the ensemble of polymers and oligomers investigated in Ref. (9), the mean results:

FIG. 1. Surface tension g as a function of temperature for two homopolymers and a copolymer. Circles, expt. (23, 24, 28); crosses and lines, computed, Eq. [2].

universality of these characteristic ratios seen earlier seems to exist for larger chainlengths. To make predictive estimates of surface tension, utilizing Eq. [2], we develop relations between g 0 and T c with the scaling parameters of the EOS. Following previous rationalizations (9) we examine g 0 as a function of the aggregate (T*P* 2 ) 1/3 , and find

g 0 5 29.47 3 10 23 ~T*P* 2 ! 1/3 1 65.94, 6 # n # 17 g 0 5 27.62 3 10 23 ~T*P* 2 ! 1/3 1 65.74; 12 # n # 20.

[4-I] [4-II]

Finally, T c is, as in the polymer case, a linear function of T*, viz.:

A/ g 0 5 0.73 6 0.01; 2B/ g 0 5 1.88 6 0.13

T c 5 9.53 3 10 22 T* 1 17.15, 6 # n # 17

For n 5 17, Table 3 shows corresponding ratios of 0.76 and 2.11. On the other hand, Method II yields for n 5 20, 0.74 and 1.92. It appears then that the near

T c 5 5.17 3 10 22 T* 1 125.0, 12 # n # 20.

[5-I] [5-II]

TABLE 2 Characteristic EOS Scaling Temperatures T*, and Pressures P*, and Coefficients in Eqs. [3-I, 3-II] for n-Alkanes Method I a

Method II a

n

T*(K)

P* (MPa)

a

b

2c

T*(K)

P* (MPa)

b0

2c 0

6 7 8 9 10 11 12 13 14 15 16 17 18 20

5197 5468 5758 6112 6301 6475 6717 6942 7115 7271 7448 7539 — —

720.1 704.6 709.6 701.1 691.0 686.9 681.2 684.7 682.5 680.3 678.5 677.8 — —

16.381 16.049 15.769 15.478 15.216 14.978 14.761 14.562 14.380 14.211 14.056 13.912 — —

3.9853 4.0546 4.1139 4.1712 4.2225 4.2687 4.3106 4.3487 4.3835 4.4154 4.4448 4.4719 — —

0.0905 0.0910 0.0914 0.0918 0.0922 0.0925 0.0929 0.0931 0.0934 0.0936 0.0939 0.0941 — —

— — — — — — 10287 10749 10997 11308 11561 11827 12140 12523

— — — — — — 746.4 749.4 743.2 741.3 737.9 736.2 738.3 731.5

— — — — — — 8.930 8.829 8.952 9.211 9.145 9.220 9.464 9.436

— — — — — — 0.1407 0.1328 0.1322 0.1355 0.1302 0.1291 0.1325 0.1127

a

See text.

426

NOTE

TABLE 3 Critical Temperatures T c, Coefficients A and B, Eq. [1], and Characteristic Ratios A/^g 0& and 2B/^g 0& for n-Alkanes Method I

Method II

n

T c a(K)

A

2B

A/^ g 0 &

2B/^ g 0 &

A

2B

A/^ g 0 &

2B/^ g 0 &

6 7 8 9 10 11 12 13 14 15 16 17 18 20

508 540 569 595 619 640 659 677 695 710 725 735 750 778

35.70 37.08 37.04 37.23 37.67 38.04 38.00 38.50 38.75 38.72 38.89 39.05 — —

92.4 95.1 95.8 99.2 100.5 100.8 101.7 106.1 108.0 107.7 109.7 109.2 — —

0.681 0.697 0.708 0.710 0.723 0.729 0.729 0.740 0.748 0.750 0.754 0.755 — —

1.763 1.788 1.832 1.893 1.929 1.931 1.952 2.040 2.085 2.087 2.126 2.112 — —

— — — — — — 36.85 37.59 38.05 37.88 38.46 38.56 38.69 38.19

— — — — — — 100.2 106.1 106.5 103.8 106.4 106.0 106.7 99.3

— — — — — — 0.707 0.668 0.735 0.734 0.745 0.746 0.753 0.738

— — — — — — 1.923 1.825 2.056 2.012 2.060 2.050 2.076 1.917

a

Ref. (15).

Relations [4-I, 4-II] and [5-I, 5-II] predict the respective quantities g 0 and T c with less than one percent deviation. Substituting into Eq. [2], the maximum deviations between computed and experimental surface tension are: I: 6 2% for 8 # n # 17, , 3% for n 5 6, 7

The first yields a minor temperature dependence, i.e., a decrease of about 10% over 80 degrees, as seen in Table 4. The results at 20°C are practically identical for Methods I and II. The comparison with experimental temperature coefficients derived from a linear g 2 T dependence shows excellent agreement with the results of Eq. [6-II] and intermediate values between 20 and 100°C for Eq. [6-I].

II: 6 2% over the entire range 12 # n # 20.

POLYMERS In noting the differences in the numerical values resulting from procedures I and II, it should be recalled that these arise not only from the differences in numerical analysis, but may also involve differences in the experimental PVT sources. Moreover, the measurements of surface tension which extend over more than 100 degrees are due to several authors. Finally, we compute the temperature coefficients by means of Eq. [1], viz.: d g /dT 5 ~B/T*!~dh/dT˜ ! with the aid of Eqs. [3-I] or [3-II], that is dg 5 ~B~n!/T*!@2a~n!T˜ 1 b~n!# dT

[6-I]

dg 5 ~B~n!/T*!b 0 ~n!. dT

[6-II]

In this section we consider the results of surface tensions computed from Eq. [2] using Eqs. [15] and [18] of Ref. (9) for those polymers, whose scaling parameters are available in the literature (21, 22). In Table 5 the EOS and surface tension parameters, and the ratios of experimental to computed g’s, with the corresponding temperature ranges, are listed for a series of polymers and copolymers. We note that no parameter adjustments in the EOS have been allowed. In a majority of instances, the deviations do not exceed 3%, with others going as high as ;10%. Figure 1 shows some examples of computation and measurement (23, 24). It is of interest to estimate surface tensions of some high T g systems. In Fig. 2 are displayed g values as a function of T/T*. We note the close positions of the three highest T g polymers with polysulfone showing the smallest g. The appropriate parameters (25) are shown in Table 5. Table 6 lists the experimental and computed temperature coefficients. Good agreement is noted in most instances, but with several exceptions, in particular

TABLE 4 Temperature Coefficients of Surface Tension of n-Alkanes n

Eq. [6-I] Eq. [6-I] Eq. [6-II]

2(d g /dt) t520 2(d g /dt) t5100 2(d g /dt)

6

7

8

9

10

11

12

13

14

15

16

17

18

20

0.104 0.113 —

0.100 0.109 —

0.095 0.102 —

0.092 0.098 —

0.090 0.096 —

0.088 0.093 —

0.085 0.090 0.087

0.085 0.090 0.087

0.085 0.089 0.087

0.082 0.087 0.085

0.082 0.086 0.084

0.080 0.085 0.083

— — 0.083

— — 0.075

427

NOTE

TABLE 5 Scaling Parameters P*, T*, g 0, T c, and g(Exp)/g(calc.)

Polymer

P* (MPa)

T*(K)

T c (K)

g 0 (mN/m)

a-Polypropylene i-Polypropylene Polyisobutylene Polystyrene Polyethylene oxide Polyethylacrylate Polymethyl acrylate Nylon 6 Nylon 66 Polyepichlorohydrin Polyvinyl acetate Ethylene/vinylacetate 18 wt% Ethylene/vinylacetate 25 wt% Ethylene/vinylacetate 28 wt% Ethylene/vinylacetate 40 wt% Bisphenol-A Polycarbonate Polyarylate Phenoxy Polysulfone

604.8 517.9 681.4 713.3 907.6 739.0 843.8 643.5 748.6 844.1 942.7 679.8 666.9 699.4 738.0 992.6 994.2 1115.4 1087.6

9360 11260 11400 12680 10170 9929 10360 15290 11980 11270 9389 10670 10430 10370 10330 11802 12903 11546 12560

861 1010 1021 1121 924 905 939 1325 1066 1010 863 963 945 940 937 1052 1138 1032 1111

50.3 48.1 58.8 63.1 69.2 59.3 66.1 62.7 64.0 68.1 69.1 57.3 56.1 57.9 60.1 77.6 80.2 83.6 84.6

for PVAc and its copolymers. However, here the experimental data do not exhibit regular variations with composition. Finally, it is tempting to estimate the compositional dependence of g in polymer mixtures, based once more on Eq. [2] and Eqs. [15] and [18] of Ref. (9) with the appropriate EOS parameters and disregarding any local interfacial contributions. Such results are exhibited in Fig. 3 for the polystyrene-poly(2,6dimethyl 1,4-phynelene ether) pair. The EOS determined by Zoller and Hoehn (26), and analyzed by Jain et al. (27) provides the composition dependence of T* and P*. Essentially linear variations with composition expressed as weight (circles) or mole (crosses) fraction are noted at all temperatures.

g ~Exp! g ~Calc! 1.02–1.03 0.89–0.86 0.86–0.82 0.88–0.85 0.99–1.01 1.13 0.97 1.09 1.13 0.97 0.89–0.93 0.93–0.96 0.98–1.00 0.92–0.98 0.92–0.95 — — — —

T-range °C 120–190 190–230 50–180 178–238 80–200 140–180 140–180 265–285 270–300 25°C 70–180 20–170 20–180 120–180 120–180 — — — —

The general approach formulated earlier to relate surface and bulk thermodynamic properties by means of a theoretical free volume function is now

found to be also applicable to small chain-molecular fluids, as represented by the n-alkane series. That is, the equations for the Guggenheim parameters, g 0 and T c (9), stand with altered chainlength dependence. A near universality of the characteristic ratios A/ g 0 and B/ g 0 is maintained. Excellent agreement between experiment and computation in respect to both surface tension and its temperature derivative ensues. These findings suggest a consideration of nonlinear hydrocarbons. Evaluations of EOS data and resulting free volume functions have already been obtained (19, 20). The application of earlier relations (9) to a number of high polymers predicts in most cases surface tensions and temperature coefficients successfully. The extension to other specific polymers and to polymer mixtures with known EOS information is now open and suggests experimental efforts. We believe the positive findings obtained rest on the quantitative success of the lattice-hole theory which does not require parameter adjustments. The formulation of a theory consistent with lattice-hole theory in the bulk still remains.

FIG. 2. Predicted surface tension g as a function of scaled temperature T/T* for some high T g polymers: (1) Phenoxy; (2) polyarylate; (3) bisphenol-A polycarbonate; (4) polysulfone.

FIG. 3. Predicted surface tension g (circles) of polystyrene-poly(2,6 dimethyl phenylene oxide) mixtures vs weight fraction of PPO at three temperatures. Crosses, linear dependence on mole fraction; lines for guidance.

CONCLUSIONS

428

NOTE

TABLE 6 Temperature Coefficients of Surface Tension (2dg/dt)

Polymer

LPE

a-PP

i-PP

PIB

PS

PEO

PEA

PMA

Nylon 66

PVAc

EVA 18

EVA 25

EVA 28

EVA 40

PDMS

Experiment Computation

0.067 0.070

0.059 0.057

0.058 0.050

0.064 0.063

0.067 0.063

0.076 0.081

0.077 0.070

0.077 0.083

0.065 0.063

0.066 0.085

0.054 0.065

0.067 0.065

0.037 0.067

0.047 0.068

0.042 0.054

REFERENCES 1. Corner, J., Trans. Faraday Soc. 44, 1036 (1948). 2. Prigogine, I., and Saraga, L., J. Chim. Phys. 49, 399 (1952). 3. Van Ness, K. E., and Couchman, P. R., J. Colloid Interface Sci. 182, 110 (1996). 4. Roe, R. J., Proc. Natl. Acad. Sci. U.S.A. 56, 819 (1966). 5. Poser, C. I., and Sanchez, I. C., J. Colloid Interface Sci. 69, 539 (1979). 6. Dee, G. T., and Sauer, B. B., J. Colloid Interface Sci. 152, 85 (1992). 7. Dee, G. T., and Sauer, B. B., Polymer 36, 1673 (1995). 8. Sanchez, I. C., J. Chem. Phys. 79, 405 (1983). 9. Carri, G. A., and Simha, R., J. Colloid Interface Sci. 178, 483 (1996). 10. Guggenheim, E. A., “Thermodynamics,” p. 199. North Holland, Amsterdam, 1959. 11. Dee, G. T., and Walsh, D. J., Macromolecules 21, 815 (1988). 12. Dee, G. T., Ougizawa, T., and Walsh, D. J., Polymer 33, 3462 (1992). 13. Utracki, L. A., private communication. 14. Simha, R., and Somcynsky, T., Macromolecules 2, 342 (1969). 15. Vargaftik, N. B., “Handbook of Physical Properties of Liquids and Gases,” 2nd ed. Wiley, New York, 1975. 16. Jain, R. K., and Simha, R., J. Chem. Phys. 70, 2792, 5329 (1979). 17. Jain, R. K., and Simha, R., Macromolecules 13, 1501 (1980). 18. Jain, R. K., and Simha, R., Ber. Bunsenges. Phys. Chem. 85, 626 (1981). 19. Yahsi, U., Ph.D. Thesis, Case Western Reserve University, 1994. 20. Simha, R., and Yahsi, U., Faraday Trans. 91, 2443 (1995). 21. Hartmann, B., Simha, R., and Berger, A. E., J. Appl. Polym. Sci. 43, 983 (1991).

22. Rodgers, P. A., J. Appl. Polym. Sci. 48, 1061 (1993). 23. Wu, S., “Polymer Interface and Adhesion.” Marcel Dekker, New York, 1982. 24. Wu, S., Macromol. Sci.-Revs. Macromol. Chem. C10(1), 1 (1974). 25. Zoller, P., J. Polym. Sci. Polym. Phys. Ed. 20, 1453 (1982). 26. Zoller, P., and Hoehn, H. H., J. Polym. Sci. Polym. Phys. Ed. 20, 1385 (1982). 27. Simha, R., and Jain, R. K., J. Polym. Sci. Polym. Phys. Ed. 20, 1399 (1982). 28. Roe, R. J., J. Phys. Chem. 72, 2013 (1968). Raj K. Jain 1 Robert Simha 2 Department of Macromolecular Science Case Western Reserve University Cleveland, Ohio 44106-7202 Received July 27, 1998; accepted May 19, 1999

1 Permanent address: Rajdhani College, University of Delhi, Raja Garden, New Delhi 110015, India. 2 To whom correspondence should be addressed.