Second and third virial coefficients of low-molecular-weight polystyrene with a benzyl end in toluene

Abstract

Light-scattering measurements were taken on six samples of polystyrene (PS) with a benzyl group at one end of the chain (benzyl-PS) in toluene at 15 °C. The second and third virial coefficients (A₂ and A₃, respectively) were determined as functions of weight-average molecular weight, M_w, which ranges from 6.0 × 10² to 1.2 × 10⁴. It was found that values of A₂ for benzyl-PS were systematically smaller than those for PS with a butyl group at one end of the chain (butyl-PS) and that the difference becomes larger with decreasing M_w. From the A₂ data obtained, the excess binary-cluster integral between the butyl end and the middle segments was estimated to be about three times larger than the binary-cluster integral between the benzyl end and the middle segments, where ‘excess’ means the difference from the binary-cluster integral between middle segments. The A₃ values for benzyl-PS for M_w<10⁴ were also smaller than those for butyl-PS. If we assume a term independent of molecular weight in A₃, the excess ternary-cluster integral among one butyl end and two middle segments was estimated to be about five times larger than that among one benzyl end and two middle segments.

Introduction

Solution properties of long flexible polymers are discussed on the basis of the two-parameter theory.^{1, 2} However, this theory cannot describe properties in low-molecular-weight regions, in which effects from stiffness and/or local conformations of the chain, chain-end groups and three-segment interactions become essential.³

The importance of the effects of chain stiffness on A₂ was discussed by Yamakawa.⁴ Einaga et al.⁵ later showed that the effects of chain ends, as well as chain stiffness, must be considered to obtain a quantitative agreement between the theoretical and observed values of A₂ for polystyrene (PS) in toluene (a good solvent) for M_w<10⁴, where M_w denotes the weight-average molecular weight. The PS samples used by Einaga et al.⁵ had a butyl group at one end of the chain. If we perform a similar study on PS samples with different chain ends, different results may be obtained.

It is known that data for A₂ in cyclohexane solutions of PS with a butyl group (butyl-PS) versus PS with a benzyl group (benzyl-PS) at one end of the chain show very different tendencies for M_w<10⁴ at the theta point (34.5 °C), where A₂ for sufficiently large M_w vanishes. The positive A₂ of butyl-PS increasing with decreasing M_w is attributed to effects from the chain ends.⁴ On the other hand, values of A₂ for benzyl-PS became negative and decreased with decreasing M_w.⁶ It was considered that the negative values of A₂ occurred because of the effects of the three-segment interactions and that effects from chain ends were negligibly small. From these results on the theta solvent system, we may expect to reduce the chain-end effect on A₂ for a good solvent system by changing the chain end from a butyl group to a benzyl group.

Compared with A₂, the number of studies on the third virial coefficient, A₃, of polymer solutions is much smaller because of the difficulty of obtaining precise values. Sato et al.⁷ overcame this problem by applying the Bawn plot⁸ (see Results and Discussion section), which enables a separate determination of A₂ and A₃. Nakamura et al.⁹ applied this plot to PS+benzene solutions to obtain systematic data on A₂ and A₃ as functions of M_w. They showed that the reduced third virial coefficient, g, defined by A₃/A₂²M_w, is an increasing function of M_w for M_w≳10⁵. This is consistent with the prediction from the two-parameter theory, except that the values of g were almost constant for 10⁴≲M_w≲10⁵. Norisuye et al.¹⁰ examined g in the low-molecular-weight range and found that three-segment interaction has a more important role than chain stiffness. Later, Osa et al.¹¹ reached essentially the same conclusion from a study on poly (α-methylstyrene) in toluene. In these studies, only data for M_w>10⁴ were discussed without considering the effects from chain ends. Because the effect of the chain end on A₃ of butyl-PS in cyclohexane is known to be remarkable for M_w≲10⁴,¹² the same effect on A₃ should be considered for good solvent systems.

In this study, we performed light-scattering measurements to determine A₂ and A₃ for benzyl-PS, with M_w ranging from 6.0 × 10² to 1.2 × 10⁴ in toluene at 15 °C. This enabled us to study the effects from the chain end and chain stiffness by comparing the results with the data for butyl-PS.⁵

Experimental procedure

Benzyl-PS samples (PS600, PS700, PS4800, PS8300 and PS12000) were obtained by anionic polymerization of styrene with benzyllithium as the initiator¹³ in an argon atmosphere. These samples, except the two lowest molecular-weight samples, were fractionally precipitated with benzene as the solvent and methanol as the precipitant. Sample PS1900 was prepared in the previous study.⁶ The number-average to weight-average molecular weight ratio, M_w/M_n, was determined by gel-permeation chromatography. Values were less than 1.06, as shown in Table 1. The two samples, PS600 and PS700, were obtained by fractionation of low-molecular-weight benzyl-PS by gel-permeation chromatography with chloroform as the eluent, and they were then reprecipitated from acetone solutions to water. These liquid samples were dried at 50 °C in a rotary evaporator for 2 weeks. Existence of the benzyl chain end was confirmed by ¹H-NMR and MALDI-TOF mass spectroscopy.

Table 1 Properties of benzyl-end polystyrene samples in toluene at 15 °C

Each test solution was optically clarified by filtration through a double layer of Teflon filters, with 0.1 μm pores, into a cylindrical light-scattering cell. The cell had been rinsed with refluxing acetone for more than 1.5 h and dried overnight in a desiccator.

The polymer mass concentration, c, of each solution was calculated from the gravimetrically determined polymer weight fraction, w, multiplied by the solution density ρ, which was calculated from ρ=ρ₀[1+(1−$\overline{v}$ ρ₀)w]. Here, ρ₀ and $\overline{v}$ denote the solvent density and the specific volume of the solute, respectively. The specific volumes for PS600 and PS700 were determined from density measurements of the solvent and the solutions by an Anton–Paar densitometer, DMA-5000 (Anton–Paar, Graz, Austria), as shown in Table 1. The value for the other samples was assumed to be 0.91 cm³ g⁻¹.¹⁴

Light-scattering measurements were taken on a Fica 50 light-scattering photometer (Fica, Saint Denis, France) with vertically polarized light with a wavelength of 436 nm in an angular range from 22.5 to 142.5° to obtain the excess Rayleigh ratio, R_θ, at the scattering angle θ. The data for each solution were extrapolated to θ=0 by plotting Kc/R_θ against sin²θ to obtain the zero-angle value Kc/R₀, where K represents the optical constant. Corrections of optical anisotropy were made for both solutions and solvent according to the following equation:

Here, R₀^* denotes the uncorrected R₀ and ρ_u is the depolarized ratio, which was determined by the method of Rubingh and Yu.¹⁵

The specific reflective index increments, ∂n/∂c, measured by a modified Schulz—Cantow-type differential refractometer, are summarized in the fifth column of Table 1.

Results and Discussion

Data analysis

The scattering intensities, extrapolated to θ=0, were obtained as a function of c, for each polymer sample. Except PS600 and PS700, they were analyzed using the Bawn plot,⁸ that is, S(c₁,c₂), as defined by the following equation plotted against c₁+c₂:⁷

Here, means (Kc/R₀) at c=c_i (i=1, 2). Figure 1 shows the Bawn plots for the samples indicated. Each plot can be fitted to a straight line. The slope and intercept at c₁+c₂=0 are equated to 3A₃ and 2A₂, respectively, according to Equation (1). We tried to apply the same plot to the data for PS600 and PS700, but we could not draw a line uniquely for each plot because of the considerable scattering of data points. For these samples, we applied the Berry plot,¹⁶ that is, (Kc/R₀)^1/2 against c, as is shown in Figure 2. The slope and the intercept at c=0 of the fitted straight line for each plot are equated to M_w^1/2A₂ and M_w^−1/2, respectively. The values for A₂ and A₃, along with M_w, are summarized in Table 1.

Figure 1

Bawn plots for benzyl-end polystyrene samples in toluene at 15 °C.

Figure 2

Square-root plots for benzyl-end polystyrene samples in toluene at 15 °C.

Second virial coefficient

The dependence of A₂ on molecular weight is illustrated in Figure 3. The unfilled and filled circles represent the data points of A₂ for benzyl- and butyl-PS,⁵ respectively, in toluene at 15 °C. The data points for benzyl-PS are systematically lower than those for butyl-PS. The difference becomes larger with decreasing M_w.

Figure 3

Dependence of A₂ on molecular weight for polystyrenes in toluene at 15.0 °C. Filled and unfilled circles represent data for butyl-end⁵ and benzyl-end polystyrenes, respectively. The dotted curve indicates theoretical values without chain-end effects. Solid and dashed curves represent theoretical values with chain-end effects.

Theoretically, A₂ is expressed by the following equation:⁴

A₂^(HW) represents the term for the binary interactions between chain segments, not including the contribution of the chain-end effects, and may be calculated from the following equation for the helical worm-like (HW) chain³ with beads or segments arrayed along the chain contour with a distance of a:

N_A, L and M denote the Avogadro constant, the contour length and molecular weight, respectively. c_∞ is expressed as³

with the stiffness parameter, λ⁻¹, and the differential geometrical curvature, κ₀, and torsion, τ₀. The excluded-volume parameter, B, is defined by

β₂ is the binary-cluster integral between middle segments. Here, the word middle means the segments excluding the ones at the chain ends. The dimensionless h in Equation (3) may be calculated from the modified Barrett equation^{4, 17} as a function of λB and λL with the scaled excluded-volume parameter (see Einaga et al.⁵).

The chain-end term, A₂^(E), may be expressed as⁴

The constants a_2,1 and a_2,2 are defined as⁴

with

M₀ is the molecular weight for one segment. β_ij is the excess binary-cluster integral between segments i and j over the binary-cluster integral between the middle segments, β₂, where the subscripts 0, 1 and 2 represent the segments at the middle, at one of the chain ends and at the other chain end, respectively. A monomer unit is regarded as a segment.

The dotted line in Figure 3 represents the calculated A₂^(HW) from Equation (3), with the following parameter set for PS in toluene:¹⁸ M_L=358 nm⁻¹, λ⁻¹=2.06 nm, λ⁻¹κ₀=3.0, λ⁻¹τ₀=6.0 and λB=0.33. M_L is the molecular weight for the unit contour length. The calculated values for M_w≲10⁵ are systematically lower than both experimental values for butyl- and benzyl-PS. The difference between theoretical and experimental values becomes larger with decreasing M_w. By adding A₂^(E) with a_2,1=2.5 cm³ g⁻¹ and a_2,2=−200 cm³ mol⁻¹,⁵ we obtain the dashed line, which closely fits the experimental data for butyl-PS. The solid line calculated with a_2,1=1.4 cm³ g⁻¹ and a_2,2=−200 cm³ mol⁻¹ also closely fits the data for benzyl-PS. From Equation (7), with these values of a_2,1, we calculate β_2,1 for butyl- and benzyl-PS to be 0.22 and 0.12 nm³, respectively.

Both ends of the benzyl-PS chain have the same chemical structure, because a polystyryl anion gives the benzyl structure when it is deactivated with a proton. Thus, we may regard β₀₁=β₀₂ in Equation (9) for benzyl-PS to obtain β₀₁=0.12 nm³ for the binary interaction between the benzyl end and middle (or PS) segments. The structure of the terminated end of butyl-PS is the same as that of benzyl-PS. As the values for β₀₁ for both polymers are the same, we obtain 0.32 nm³ for β₀₂ from Equation (9) with the above β₀₁ and β_2,1 for butyl-PS. The value of β₀₂, which represents the binary interaction between the butyl end and middle segments, is about three times larger than β₀₁ for the benzyl end.

Positive β₀₁ shows that the net binary-cluster integral between the benzyl end and middle segments is larger than the one between middle segments, β₂. These results are rather different from those for PS in cyclohexane (34.5°C), a theta solvent, in which β₀₁ is vanishingly small.⁶ In this solvent, the binary-cluster integral between the benzyl end and middle (PS) segments may be incidentally close to the one between middle segments.

Third virial coefficient

Figure 4 shows the dependence of A₃ on molecular weight, in which literature data for butyl-PS in toluene at 15 °C¹⁹ and those for butyl-PS in benzene at 25 °C^{7, 9} are included. The values of A₃ for benzyl-PS from this study are slightly lower than those for butyl-PS.

Figure 4

Dependence of A₃ on molecular weight for polystyrenes in good solvents. Filled and unfilled circles represent data for butyl-end (no pip, in toluene at 15 °C;¹⁹ pip down⁷ and right,⁹ in benzene at 25 °C) and benzyl-end polystyrenes, respectively. Dotted, dot-dashed and solid curves represent theoretical values for , respectively (with β₃=4.7 × 10⁻³ nm⁶).

The third virial coefficient for the linear chain may be expressed as¹²

The first and second terms on the right-hand side of the above equation correspond to the contribution of the binary- and ternary-segment interactions, respectively. We note that the remaining part, which includes cross terms of β₂ and β₃,²⁰ is ignored here because it is anticipated to be negligibly small.

The first term on the right-hand side of Equation (11) may be expressed as

g₂ may be calculated from the following equation:^{10, 11}

z̄ is defined as

with

The radius expansion factor, α_S, may be calculated from the modified Domb–Barrett equation:²¹

with

where K(λL) is a known function.^{3, 22}

The second term on the right-hand side of Equation (11) may be expressed as¹²

where the first term on the right-hand side represents the contribution from the interaction among three middle segments. The other term is for the interaction among two middle segments and one chain-end segment.

The zeroth-order contribution to the former term can be expressed as^{1, 23}

with the ternary-cluster integral, β₃, among middle segments. If we ignore the higher term of the order of M⁻², we may express the chain-end term as¹²

Here, a_3,1 is given by

with

β_00i denotes the excess ternary-cluster integrals among two middle segments (denoted by 0) and one end segment i (i=1,2) over that among three middle segments β₃.

The dotted and dot-dashed lines in Figure 4 show and respectively, according to the above equations, with the parameter set for PS in toluene given in the previous section. Here, we take β₃=4.7 × 10⁻³ nm⁶, which was obtained from the value of A₃ of sufficiently high-molecular-weight PS in cyclohexane at the theta point.²⁴ Both lines cannot explain the experimental data for M_w≲10⁵. By adding with a_3,1=8 cm⁶ g⁻² to the dot-dashed line, we obtain the solid line, which is somewhat closer to the data points. However, the agreement is limited to a restricted molecular weight range.

A better fit can be obtained if we use β₃=1.8 × 10⁻² nm⁶, a much larger β₃ than the one used above, as shown by the dot-dashed line in Figure 5 for Use of such a large β₃ was suggested by Koyama and Sato.¹⁹ The solid and dashed lines in Figure 5 were obtained by adding with a_3,1=1.0 and 3.0 cm⁶ g⁻², respectively, to the dot-dashed line. This led to satisfactory fits to the data points for benzyl- and butyl-PS, respectively. From these values of a_3,1, we calculate β_3,1 to be 0.015 and 0.045 nm⁶ for the benzyl- and butyl-PS, respectively. The former β_3,1 gives β₀₀₁=0.015 nm⁶ for the benzyl end. The latter, with this β₀₀₁, gives β₀₀₂=0.075 nm⁶ for the butyl end. These results show that β₀₀₂ is about five times larger than β₀₀₁.

Figure 5

Dependence of A₃ on molecular weight for polystyrenes in good solvents. Symbols and dotted and dashed curves have the same meaning as those in Figure 4. Solid and dashed lines represent theoretical values for benzyl- and butyl-end polystyrenes, respectively (with β₃=1.8 × 10⁻² nm⁶).

The β₃ value used for Figure 5 suggests that β₃ in toluene at 15 °C is about four times larger than that in cyclohexane at 34.5°C. This seems rather odd because calculations on rigid spheres with the Lennard–Jones potential show that β₃ at the Boyle point (corresponding to the theta point) is larger than the one at higher temperatures.²⁵ Furthermore, if we calculate β₃/β₂² from the above β₃ and β₂=0.041 nm³, as estimated from λB=0.33, we obtain β₃/β₂²=11, which is much larger than the rigid-sphere value of 0.625. However, the β₃ term of Equation (19) only adds a constant to A₃. It is likely that we overlooked a contribution independent of molecular weight, although we do not know what it is.

Conclusion

By combining the data for A₂ and A₃ for low-molecular-weight butyl- and benzyl-PS, we estimated the excess binary-cluster integrals (β₀₁ and β₀₂) between chain-end and middle segments, as well as the excess ternary-cluster integrals (β₀₀₁ and β₀₀₂) among one chain-end segment and two middle segments. The conclusions obtained from the study are summarized as follows: (1) β₀₂ for the interaction between the butyl end and middle segments is about three times larger than β₀₁ for the interaction between the benzyl end and middle segments. (2) The data analysis of A₃ suggested a contribution that is independent of molecular weight by some effect. (3) If we assume such a contribution, β₀₀₂ for the interaction among one butyl end and two middle segments is estimated to be about five times larger than β₀₀₁ for the interaction among one benzyl end and two middle segments.