Excluded-Volume Effects

Abstract

This chapter deals with the theory of the excluded-volume effects in dilute solution, such as various kinds of expansion factors and the second and third virial coefficients, developed on the basis of the perturbed HW chain which enables us to take account of both effects of excluded volume and chain stiffness. Necessarily, the derived theory is no longer the two-parameter (TP) theory [1], but it may give an explanation of experimental results [2] obtained in this field since the late 1970s, which all indicate that the TP theory breaks down. There are also some causes other than chain stiffness that lead to its breakdown. On the experimental side, it has for long been a difficult task to determine accurately the expansion factors since it is impossible to determine directly unperturbed chain dimensions in good solvents. However, this has proved possible by extending the measurement range to the oligomer region where the excluded-volume effect disappears. Thus an extensive comparison of the new non-TP theory with experiment is made mainly using such experimental data recently obtained for several flexible polymers. As for semiflexible polymers with small excluded volume, some remarks are made without a detailed analysis.

Appendices

Appendix 1: Mean-Square Electric Dipole Moment

Experimentally, Marchal and Benoit [109] first showed that there is no excluded-volume effect on the mean-square electric dipole moment $\langle \mu ^{2}\rangle$ for the chain having type-B (perpendicular) dipoles like polyoxyethyleneglycol and diethoxy polyethyleneglycol. On the theoretical side, Nagai and Ishikawa [110] and subsequently Doi [111] supported this conclusion on the basis of the Gaussian chain, that is, α _μ  = 1 if $\langle \mathbf{R} \cdot \boldsymbol{\mu }\rangle _{0} = 0$ with R the end-to-end vector distance and $\boldsymbol{\mu }$ the instantaneous (total) electric dipole moment vector. However, Mattice and Carpenter [112] have reported a Monte Carlo result in contradiction to the above conclusion on the basis of the RIS model; that is, α _μ is not equal to unity for the type-B chain of finite length, and moreover, it does not become unity even in the limit of $L \rightarrow \infty$. Mansfield [113] has then clarified that their result is due to the non-Gaussian nature of the chain, although not completely molecular-theoretically.

Thus, in this appendix we evaluate α _μ  ² (only its first-order perturbation coefficient) on the basis of the HW chain [114]. All lengths are measured in units of $\lambda ^{-1}$ as usual, and the same notation as that in Sect. 5.4.1 is used. By the use of Eq. (5.155), $\langle \mu ^{2}\rangle$ may be written in the form

$$\displaystyle{ \langle \mu ^{2}\rangle =\int _{ 0}^{L}\int _{ 0}^{L}{\bigl \langle\tilde{\mathbf{m}}(t_{ 1}) \cdot \tilde{\mathbf{m}}(t_{2})\bigr \rangle}dt_{1}dt_{2}\,. }$$

(8.185)

Corresponding to Eq. (8.4) with Eqs. (8.9) and (8.10) for α _R  ², the first-order perturbation theory of α _μ  ² may then be given by

$$\displaystyle{ \alpha _{\mu }^{\ 2} = 1 + K_{\mu }(L)\,z + \cdots }$$

(8.186)

with

$$\displaystyle{ K_{\mu }(L) = \frac{F_{\mu }(L)} {L^{1/2}\langle \mu ^{2}\rangle _{0}}\,, }$$

(8.187)

where

$$\displaystyle\begin{array}{rcl} F_{\mu }(L)& =& {\biggl (\frac{2\pi c_{\infty }} {3} \biggr )}^{3/2}\int _{ 0}^{L}ds_{ 1}\int _{s_{1}}^{L}ds_{ 2}\biggl \{G(\mathbf{0};s)\langle \mu ^{2}\rangle _{ 0} \\ & & -\int _{0}^{L}dt_{ 1}\int _{0}^{L}dt_{ 2}\int \bigl [\tilde{\mathbf{m}}(t_{1}) \cdot \tilde{\mathbf{m}}(t_{2})\bigl ]P_{0}(\Omega _{1},\Omega _{2},\mathbf{0}_{s_{1}s_{2}};L)d\Omega _{1}d\Omega _{2}\biggr \} \\ & & {}\end{array}$$

(8.188)

with $\Omega _{i} = \Omega (t_{i})$ (i = 1, 2).

The asymptotic solution in the limit of $L \rightarrow \infty$ is then found analytically to be

$$\displaystyle{ \langle \mu ^{2}\rangle _{ 0} ={\biggl [ \frac{4m^{2} + (\kappa _{0}m_{\eta } +\tau _{0}m_{\zeta })^{2}} {4 +\kappa _{ 0}^{\ 2} +\tau _{ 0}^{\ 2}} \biggr ]}L + \mathcal{O}(L^{0})\,, }$$

(8.189)

$$\displaystyle\begin{array}{rcl} K_{\mu }(L) = \frac{4} {3}{\biggl [ \frac{4 +\tau _{ 0}^{\ 2}} {4m^{2} + (\kappa _{0}m_{\eta } +\tau _{0}m_{\zeta })^{2}}\biggr ]}{\biggl ( \frac{\kappa _{0}\tau _{0}m_{\eta }} {4 + \tau _{0}^{2}} + m_{\zeta }\biggr )}^{2} + \mathcal{O}(L^{-1/2})\,.& & \\ & &{}\end{array}$$

(8.190)

For the HW chain having type-A (parallel) dipoles ($m_{\xi } = m_{\eta } = 0$), in the coil limit K _μ (L) is equal to 4/3, and therefore α _μ  = α _R , as seen from Eq. (8.190). For the HW chain having type-B dipoles (m _ζ  = 0), Eq. (8.190) reduces to

$$\displaystyle{ \qquad K_{\mu }(L) = \frac{4\kappa _{0}^{\ 2}\tau _{0}^{\ 2}m_{\eta }^{\ 2}} {3(4m^{2} +\kappa _{ 0}^{\ 2}m_{\eta }^{\ 2})(4 +\tau _{ 0}^{\ 2})} + \mathcal{O}(L^{-1/2})\qquad (\mathrm{B})\,. }$$

(8.191)

It is seen from Eq. (8.191) that K _μ ≠ 0 if m _η ≠ 0 and κ ₀ τ ₀ ≠ 0, so that α _μ then becomes infinitely large in the limit of $L \rightarrow \infty$. Such dependence of α _μ on L has not been pointed out by Mattice and Carpenter [112] and by Mansfield [113]. Further, this does not conflict with the above-mentioned result [110, 111] for the Gaussian chain since the HW chain does not necessarily satisfy the condition $\langle \mathbf{R} \cdot \boldsymbol{\mu }\rangle _{0} = 0$ even in the case of perpendicular dipoles [114]. It must also be noted that α _μ may possibly become a constant different from unity because of the term of order $L^{-1/2}$ in K _μ (L) if κ ₀ τ ₀ ≠ 0 for the type-B chain. This corresponds to the case pointed out by Mattice and Carpenter and by Mansfield.

Appendix 2: Determination of the Virial Coefficients for Oligomers

For an accurate experimental determination of the (osmotic) second and third virial coefficients A ₂ and A ₃ for oligomers, light scattering measurements are preferable. Then, however, measurements must be carried out generally for optically anisotropic and rather concentrated solutions, and necessarily several problems are encountered. In this appendix we resolve them and present a procedure suitable for the present purpose [115].

We consider a binary solution which in general is optically anisotropic and not necessarily dilute. Let R _Uv ^∗ be the reduced intensity of unpolarized scattered light for vertically polarized incident light, and let $R_{\theta }^{{\ast}}$ be the Rayleigh ratio, where the asterisk indicates the scattering from anisotropic scatterers, it being dropped for the isotropic scattering. The (isotropic) Rayleigh ratio $R_{\theta =0}$ at vanishing scattering angle Θ, which is the first desired quantity, is obtained from

$$\displaystyle{ R_{\theta =0} ={\bigl ( 1 -\tfrac{7} {6}\rho _{\mathrm{u}}\bigr )}R_{\mathrm{Uv}}^{{\ast}}\,, }$$

(8.192)

or

$$\displaystyle{ R_{\theta =0} = (1 +\rho _{\mathrm{u}})^{-1}{\bigl (1 -\tfrac{7} {6}\rho _{\mathrm{u}}\bigr )}R_{\theta =\pi /2}^{{\ast}} }$$

(8.193)

with the observed R _Uv ^∗ or $R_{\theta =\pi /2}^{{\ast}}$, where ρ _u is the depolarization ratio as defined as the ratio I _Hu∕I _Vu of the horizontal to vertical component of the scattered intensity at $\theta =\pi /2$ for unpolarized incident light. ρ _u may be determined from [116]

$$\displaystyle{ (1 +\cos ^{2}\theta ) \frac{R_{\theta }^{{\ast}}} {R_{\theta =\pi /2}^{{\ast}}} = 1 +{\biggl ( \frac{1 -\rho _{\mathrm{u}}} {1 +\rho _{\mathrm{u}}}\biggr )}\cos ^{2}\theta }$$

(8.194)

with the observed $R_{\theta }^{{\ast}}$. Note that these equations can readily be derived from the basic equations for I _fi in Sect. 5.3.2

Now, according to the fluctuation theory [1, 117, 118], $R_{\theta =0}$ may be written in the form

$$\displaystyle{ R_{\theta =0} = R_{\mathrm{d}} +\varDelta R_{\theta =0} }$$

(8.195)

with R _d and $\varDelta R_{\theta =0}$ being the density scattering (the Einstein–Smoluchowski term) and the composition scattering, respectively, and given by

$$\displaystyle{ R_{\mathrm{d}} = \frac{2\pi ^{2}\tilde{n}^{2}k_{\mathrm{B}}T} {\lambda _{0}^{\ 4}\kappa _{T}} {\biggl (\frac{\partial \tilde{n}} {\partial p}\biggr )}_{T,m}^{2}\,, }$$

(8.196)

$$\displaystyle{ \qquad \qquad \varDelta R_{\theta =0} = -\frac{2\pi ^{2}\tilde{n}^{2}k_{\mathrm{B}}TV _{0}c} {\lambda _{0}^{\ 4}} {\biggl (\frac{\partial \tilde{n}} {\partial c} \biggr )}_{T,p}^{2}\bigg/{\biggl (\frac{\partial \mu _{0}} {\partial c}\biggr )}_{T,p}\,, }$$

(8.197)

where $\lambda _{0}$ is the wavelength of the incident light in vacuum, κ _T the isothermal compressibility of the solution, $\tilde{n}$ the refractive index of the solution, p the pressure, m the ratio of the solute to solvent mass, V ₀ the partial molecular volume of the solvent, c the mass concentration of the solution, and μ ₀ the chemical potential of the solvent. We note that the molecular-theoretical basis of the term R _d has been given correctly by Fixman [119], and that the multiple scattering theory developed by Bullough [120] is in error [115].

We first rewrite Eq. (8.197). Under the osmotic condition, the chemical potential μ ₀ ⁰(T, p) of the pure solvent is equated to μ ₀(T, p +Π, c) with Π the osmotic pressure, so that we have

$$\displaystyle\begin{array}{rcl} \mu _{0}^{0}(T,p)& =& \mu _{ 0}(T,p,c) +{\biggl ( \frac{\partial \mu _{0}} {\partial p}\biggr )}_{T,c}\varPi \\ & & +\frac{1} {2}{\biggl (\frac{\partial ^{2}\mu _{0}} {\partial p^{2}}\biggr )}_{T,c}\varPi ^{2} + \frac{1} {3}{\biggl (\frac{\partial ^{3}\mu _{0}} {\partial p^{3}}\biggr )}_{T,c}\varPi ^{3} + \cdots \,.{}\end{array}$$

(8.198)

Differentiation of both sides of Eq. (8.198) with respect to c at constant T and p leads to

$$\displaystyle\begin{array}{rcl} {\biggl (\frac{\partial \mu _{0}} {\partial c}\biggr )}_{T,p}& =& -V _{0}{\biggl ( \frac{\partial \varPi } {\partial c}\biggr )}_{T,p} -\varPi {\biggl (\frac{\partial V _{0}} {\partial c} \biggr )}_{T,p} \\ & & -\frac{1} {2}\varPi ^{2}{\biggl (\frac{\partial ^{2}V _{ 0}} {\partial p\partial c} \biggr )}_{T} -\varPi {\biggl (\frac{\partial V _{0}} {\partial p} \biggr )}_{T,c}{\biggl ( \frac{\partial \varPi } {\partial c}\biggr )}_{T,p} \\ & & -\frac{1} {3}\varPi ^{3}{\biggl ( \frac{\partial ^{3}V _{ 0}} {\partial p^{2}\partial c}\biggr )}_{T} -\varPi ^{2}{\biggl (\frac{\partial ^{2}V _{ 0}} {\partial p^{2}} \biggr )}_{T,c}{\biggl ( \frac{\partial \varPi } {\partial c}\biggr )}_{T,p} + \cdots \,,{}\end{array}$$

(8.199)

where we have used the relation $(\partial \mu _{0}/\partial p)_{T,c} = V _{0}$. In general, Π and V ₀ may be expanded in powers of c as follows,

$$\displaystyle{ \frac{\varPi } {RT} = \frac{1} {M}c + A_{2}c^{2} + A_{ 3}c^{3} + \cdots \,, }$$

(8.200)

$$\displaystyle{ V _{0} = V _{0}^{0}{\biggl [1 -\frac{1} {2}{\biggl (\frac{\partial v_{1}} {\partial c} \biggr )}_{T,p,0}c^{2} + \cdots \,\biggr ]}\,, }$$

(8.201)

where R is the gas constant, V ₀ ⁰ the molecular volume of the pure solvent, v ₁ the partial specific volume of the solute, and the subscript 0 on the derivative indicates its value at c = 0.

Substitution of Eq. (8.199) with Eqs. (8.200) and (8.201) into Eq. (8.197) leads to

$$\displaystyle{ \frac{Kc} {\varDelta R_{\theta =0}} = \frac{1} {M} + 2A_{2}^{{\prime}}c + 3A_{ 3}^{{\prime}}c^{2} + \cdots }$$

(8.202)

with

$$\displaystyle{ K = \frac{2\pi ^{2}\tilde{n}^{2}} {N_{\mathrm{A}}\lambda _{0}^{\ 4}}{\biggl (\frac{\partial \tilde{n}} {\partial c} \biggr )}_{T,p}^{2}\,, }$$

(8.203)

$$\displaystyle{ A_{2} = A_{2}^{{\prime}} + \frac{RT\kappa _{T,0}} {2M^{2}} \,, }$$

(8.204)

$$\displaystyle\begin{array}{rcl} A_{3}& =& A_{3}^{{\prime}} + \frac{1} {3M}{\biggl (\frac{\partial v_{1}} {\partial c} \biggr )}_{T,p,0} + \frac{RT\kappa _{T,0}A_{2}} {M} + \frac{RT} {2M^{2}}{\biggl (\frac{\partial \kappa _{T}} {\partial c} \biggr )}_{T,0} \\ & & +\frac{(RT)^{2}} {3M^{3}} {\biggl [\kappa _{T,0}^{2} -{\biggl (\frac{\partial \kappa _{T}} {\partial p}\biggr )}_{T,0}\biggr ]}\,, {}\end{array}$$

(8.205)

where κ _T, 0 is the isothermal compressibility of the pure solvent. Thus the desired virial coefficients A ₂ and A ₃ may be obtained from Eqs. (8.204) and (8.205) with the observed light-scattering virial coefficients $A_{2}^{{\prime}}$ and $A_{3}^{{\prime}}$, which are different from the former except for large M. We note that Eq. (8.204) is equivalent to a relation derived by Casassa and Eisenberg [121].

Next we consider the problem of determining R _d at finite concentrations, although indirectly. For this purpose, we adopt the Lorentz–Lorenz relation between $\tilde{n}$ and the solution density ρ _w [122],

$$\displaystyle{ \frac{\tilde{n}^{2} - 1} {\tilde{n}^{2} + 1} = \mathrm{const.}\,\rho _{\mathrm{w}}\,, }$$

(8.206)

where we assume that the proportionality constant is independent of p. Equation (8.206) has been shown to be the best of such relations [115]. Differentiation of both sides of Eq. (8.206) with respect to p leads to

$$\displaystyle{ \kappa _{T}^{-1}{\biggl (\frac{\partial \tilde{n}} {\partial p}\biggr )}_{T,m} = \frac{(\tilde{n}^{2} - 1)(\tilde{n}^{2} + 2)} {6\tilde{n}} \,. }$$

(8.207)

Substituting Eq. (8.207) into Eq. (8.196), we obtain

$$\displaystyle{ R_{\mathrm{d}} = \frac{\kappa _{T}(\tilde{n}^{2} - 1)^{2}(\tilde{n}^{2} + 2)^{2}} {\kappa _{T,0}(\tilde{n}_{0}^{\ 2} - 1)^{2}(\tilde{n}_{0}^{\ 2} + 2)^{2}}R_{\mathrm{d,0}}\,, }$$

(8.208)

where $\tilde{n}_{0}$ and R _d,0 are the values of $\tilde{n}$ and R _d for the pure solvent, respectively.

Thus we may calculate R _d from Eq. (8.208) with the observed R _d,0, and then determine $\varDelta R_{\theta =0}$ from Eq. (8.195) with this R _d and the observed $R_{\theta =0}$. Finally, we may determine M, A ₂, and A ₃ from Eq. (8.202) with Eqs. (8.203)–(8.205) by the use of the Berry square-root plot [123] or the Zimm plot [124] and also the Bawn plot [72, 73]. For the evaluation of the optical constant K given by Eq. (8.203), note that we must use values of $\tilde{n}$ and $(\partial \tilde{n}/\partial c)_{T,p}$ at finite concentrations c. For example, the results obtained for toluene (solute) in cyclohexane (solvent) at 25.0 ^∘C are M = 93 ± 4 and $A_{2} = 1.5 \times 10^{-3}$ cm³ mol/g² (with $RT\kappa _{T,0}/2M^{2} = 1.65 \times 10^{-4}$ cm³ mol/g²) [115]. (Note that the true M of toluene is 92.)