Equilibrium Properties

Abstract

This chapter presents the statistical-mechanical treatments of equilibrium conformational or static properties, such as the mean-square radius of gyration, scattering function, mean-square optical anisotropy, and mean-square electric dipole moment, of the unperturbed HW chain, including the KP wormlike chain as a special case, by an application of its chain statistics developed in Chap. 4 A comparison of theory with experiment is made with experimental data obtained for several flexible polymers in the $\varTheta$ state over a wide range of molecular weight, including the oligomer region, and also for typical semiflexible polymers (without excluded volume) in some cases. It must be noted that well-characterized samples have recently been used for measurements of dilute-solution properties of the former; they are sufficiently narrow in molecular weight distribution, and have a fixed stereochemical composition independent of the molecular weight in the case of asymmetric polymers.

Appendices

Appendix 1: Chain-Thickness Correction for the Apparent Mean-Square Radius of Gyration

Fig. 5.22

Double-logarithmic plots of $(k_{\mathrm{B}}T)^{2}A_{\mathrm{ED}}$ (in D²) against x for PBIC in CCl₄ at room temperature [10]. The solid curve represents the best-fit KPA theoretical values calculated with the values of the model parameters given in Table 5.4 along with m ₀ = 1. 25 D and ε ₁ = 2. 60

The reciprocal of the excess reduced scattered intensity $R_{\theta }$ for dilute solutions of mass concentration c may be expanded in the form [33, 84]

$$\displaystyle{ \frac{Kc} {R_{\theta }} = \frac{1} {MP_{\mathrm{s}}(k)} + 2A_{2}Q(k)c + \cdots \,, }$$

(5.205)

where K is the optical constant, M is the polymer molecular weight, P _s(k) is the scattering function as a function of the magnitude k of the scattering vector k given by Eq. (5.20), A ₂ is the second virial coefficient, and Q(k) represents the intermolecular interference. The function P _s contains effects of the spatial distribution of scatterers (electrons or hydrogen nuclei), that is, effects of chain thickness in the case of small-angle X-ray or neutron scattering. In general, it may be written in the form [1]

$$\displaystyle{ P_{\mathrm{s}}(k;L) ={\biggl \langle{\biggl |\int \rho (\mathbf{r})\exp (i\mathbf{k} \cdot \mathbf{r})d\mathbf{r}\biggr |}^{2}\biggr \rangle}\,, }$$

(5.206)

where we have explicitly indicated that P _s(k) also depends on the contour length L of the chain, $\langle \cdots \,\rangle$ denotes an equilibrium average over chain conformations, i is the imaginary unit, and ρ(r) is the excess scatterer density at vector position r and is normalized as

$$\displaystyle{ \int \rho (\mathbf{r})d\mathbf{r} = 1\,. }$$

(5.207)

In the case for which the scatterers are distributed on the chain contour, ρ(r) is given by

$$\displaystyle{ \rho (\mathbf{r}) = L^{-1}\int _{ 0}^{L}\delta {\bigl [\mathbf{r} -\mathbf{r}(s)\bigr ]}ds\,, }$$

(5.208)

where $\delta (\mathbf{r})$ is a three-dimensional Dirac delta function and r(s) is the radius vector of the contour point s (0 ≤ s ≤ L) of the chain. Then Eq. (5.206) with Eq. (5.208) gives the contour scattering function P(k; L) (without effects of chain thickness).

In the case of a cylinder model for which the scatterers are uniformly distributed within a (flexible) cylinder having a uniform cross section of area a _c whose center of mass is on the chain contour, ρ(r) is given by

$$\displaystyle{ \rho (\mathbf{r}) = (La_{\mathrm{c}})^{-1}\int _{ 0}^{L}ds\int _{\mathrm{ C}_{S}}\delta [\mathbf{r} -\mathbf{r}(s) -\bar{\mathbf{r}}_{s}]d\bar{\mathbf{r}}_{s}\,, }$$

(5.209)

where $\bar{\mathbf{r}}_{s}$ is the vector distance from the contour point s to an arbitrary point in the normal cross section at that point and the second integration is carried out over the cross section.

In the case of a touched-subbody model for which the scatterers are uniformly distributed in each of N identical touched subbodies of volume v _s whose centers of mass are on the chain contour, ρ(r) is given by

$$\displaystyle{ \rho (\mathbf{r}) = (Nv_{\mathrm{s}})^{-1}\sum _{ j=1}^{N}\int _{ \mathrm{V}_{j}}\delta (\mathbf{r} -\mathbf{r}_{j} -\bar{\mathbf{r}}_{j})d\bar{\mathbf{r}}_{j}\,, }$$

(5.210)

where r _j is the vector position of the center of mass of the jth subbody, $\bar{\mathbf{r}}_{j}$ is the vector distance from r _j to an arbitrary point within the jth subbody, and the integration is carried out within it.

The scattering function P _s may be expanded in the form

$$\displaystyle{ P_{\mathrm{s}}(k;L) = 1 -\frac{1} {3}\langle S^{2}\rangle _{ \mathrm{s}}k^{2} + \mathcal{O}(k^{4})\,. }$$

(5.211)

This is the defining equation for the apparent mean-square radius of gyration $\langle S^{2}\rangle _{\mathrm{s}}$ for the chain. It is related to the mean-square radius of gyration $\langle S^{2}\rangle$ for the chain contour by the equation

$$\displaystyle{ \langle S^{2}\rangle _{ \mathrm{s}} =\langle S^{2}\rangle + S_{\mathrm{ c}}^{\ 2}\,, }$$

(5.212)

where S _c is the radius of gyration for the cross section (cylinder model) or subbody (touched-subbody model) and is given by

$$\displaystyle\begin{array}{rcl} S_{\mathrm{c}}^{\ 2}& =& \frac{1} {8}d^{2}\ \ \ \mbox{ (cylinder)}\,,{}\end{array}$$

(5.213)

$$\displaystyle\begin{array}{rcl} S_{\mathrm{c}}^{\ 2}& =& \frac{3} {20}d_{\mathrm{b}}^{\ 2}\ \ \ \mbox{ (bead)}{}\end{array}$$

(5.214)

for the cylinder of diameter d and the sphere (bead) of diameter d _b, respectively. For the cylinder model, d has been calculated to be 9.2, 8.2, and 8.1 Å for a-PS, a-PMMA, and i-PMMA, respectively, from the partial specific volume [1, 4, 5]. For a-PS, however, the value 9.4 Å of d has been adopted in Eq. (5.212) [1].

Appendix 2: Spherical Vectors and Tensors

The spherical (irreducible) components r ^(j) (j = 0, ± 1) of a vector r = (x, y, z) are defined in terms of the Cartesian components x, y, and z by [71]

$$\displaystyle\begin{array}{rcl} & & r^{(\pm 1)} = \mp \frac{1} {\sqrt{2}}(x \pm iy)\,, \\ & & r^{(0)} = z\,. {}\end{array}$$

(5.215)

The spherical components T _l ^m (l = 0, 1, 2; m = 0, ± 1, ± 2) of a tensor $\mathbf{T} = (T_{\mu \nu })$ (μ, ν = x, y, z) are defined in terms of the Cartesian components T _{μ ν} by

$$\displaystyle\begin{array}{rcl} & & T_{0}^{0} = \frac{1} {\sqrt{3}}(T_{xx} + T_{yy} + T_{zz})\,, \\ & & T_{1}^{0} = \frac{1} {2}(T_{xy} - T_{yx})\,, \\ & & T_{1}^{\pm 1} = \mp \frac{1} {2\sqrt{2}}{\bigl [(T_{yz} - T_{zy}) \pm i(T_{zx} - T_{xz})\bigr ]}\,, \\ & & T_{2}^{0} = \frac{1} {\sqrt{6}}{\bigl [3T_{zz} - (T_{xx} + T_{yy} + T_{zz})\bigr ]}\,, \\ & & T_{2}^{\pm 1} = \mp \frac{1} {2}{\bigl [(T_{zx} + T_{xz}) \pm i(T_{zy} + T_{yz})\bigr ]}\,, \\ & & T_{2}^{\pm 2} = \frac{1} {2}{\bigl [(T_{xx} - T_{yy}) \pm i(T_{xy} + T_{yx})\bigr ]}\,.{}\end{array}$$

(5.216)

We note that the third and fifth of Eqs. (7.4.1) of [49] and all related equations are incorrect.

We have the symmetry relations

$$\displaystyle{ r^{(-j)} = (-1)^{j}r^{(j){\ast}}\,, }$$

(5.217)

$$\displaystyle{ T_{l}^{-m} = (-1)^{m}T_{ l}^{m{\ast}}\,, }$$

(5.218)

where the asterisk indicates the complex conjugate.

We have the same transformation rule as Eq. (4.264), that is,

$$\displaystyle{ \tilde{r}^{(j)} = c_{ 1}^{\ -1}\sum _{ j'=-1}^{1}\mathcal{D}_{ 1}^{jj'}(\Omega )r^{(j')}\,, }$$

(5.219)

$$\displaystyle{ \tilde{T}_{l}^{m} = c_{ l}^{\ -1}\sum _{ j=-l}^{l}\mathcal{D}_{ l}^{mj}(\Omega )T_{ l}^{j}\,, }$$

(5.220)

where the components $\tilde{r}^{(j)}$ and $\tilde{T}_{l}^{m}$ are transformed to the components r ^(j) and T _l ^m, respectively, expressed in a new Cartesian coordinate system obtained by rotation $\Omega$ of a coordinate system in which the former components are defined, $\mathcal{D}_{l}^{mj}$ are the normalized Wigner $\mathcal{D}$ functions, and c _l is given by Eq. (4.54).

Appendix 3: Proof of Nagai’s Theorem

We introduce temporarily quantities β ^(±) defined by

$$\displaystyle{ \beta ^{(\pm )} =\alpha _{\mathrm{ Vh}} \pm \alpha _{\mathrm{Hv}}\,. }$$

(5.221)

We have $\langle \tilde{\alpha }_{l_{1}}^{m_{1}},\tilde{\alpha }_{l_{ 2}}^{m_{2}}\rangle = 0$ for m ₁ ≠ m ₂, as seen from Eq. (5.90), and therefore we obtain, from Eqs. (5.87) and (5.221), the relations,

$$\displaystyle\begin{array}{rcl} \langle \alpha _{\mathrm{Vv}},\beta ^{(+)}\rangle & =& \langle \alpha _{\mathrm{ Hh}},\beta ^{(+)}\rangle =\langle \beta ^{(-)},\beta ^{(+)}\rangle = 0\,, \\ \langle \beta ^{(+)},\alpha _{\mathrm{ Vv}}\rangle & =& \langle \beta ^{(+)},\alpha _{\mathrm{ Hh}}\rangle =\langle \beta ^{(+)},\beta ^{(-)}\rangle = 0\,,{}\end{array}$$

(5.222)

and also

$$\displaystyle\begin{array}{rcl} & & \langle \alpha _{\mathrm{Vv}},\beta ^{(-)}\rangle +\langle \beta ^{(-)},\alpha _{\mathrm{ Vv}}\rangle = 0\,, \\ & & \langle \alpha _{\mathrm{Hh}},\beta ^{(-)}\rangle +\langle \beta ^{(-)},\alpha _{\mathrm{ Hh}}\rangle = 0\,.{}\end{array}$$

(5.223)

We may express $\langle \alpha _{\mathrm{fi}},\alpha _{\mathrm{fi}}\rangle$ in terms of its components $\langle \alpha _{\mathrm{Vv}}$, $\alpha _{\mathrm{Vv}}\rangle$ and so on by the use of Eq. (5.88) with β ⁽⁺⁾ and β ⁽⁻⁾ instead of $\alpha _{\mathrm{Vh}}$ and $\alpha _{\mathrm{Hv}}$. Then, if we use Eqs. (5.222) and (5.223) and change β ⁽⁺⁾ and β ⁽⁻⁾ back to $\alpha _{\mathrm{Vh}}$ and $\alpha _{\mathrm{Hv}}$, we find

$$\displaystyle\begin{array}{rcl} \langle \alpha _{\mathrm{fi}},\alpha _{\mathrm{fi}}\rangle & =& c_{\mathrm{i}}^{\ 2}c_{\mathrm{ f}}^{\ 2}\langle \alpha _{ \mathrm{Vv}},\alpha _{\mathrm{Vv}}\rangle + s_{\mathrm{i}}^{\ 2}s_{\mathrm{ f}}^{\ 2}\langle \alpha _{ \mathrm{Hh}},\alpha _{\mathrm{Hh}}\rangle + \frac{1} {2}(c_{\mathrm{i}}^{\ 2}s_{\mathrm{ f}}^{\ 2} + s_{\mathrm{ i}}^{\ 2}c_{\mathrm{ f}}^{\ 2}) \\ & & \times \bigl (\langle \alpha _{\mathrm{Vh}},\alpha _{\mathrm{Vh}}\rangle +\langle \alpha _{\mathrm{Hv}},\alpha _{\mathrm{Hv}}\rangle \bigr ) + c_{\mathrm{i}}s_{\mathrm{i}}c_{\mathrm{f}}s_{\mathrm{f}}\bigl (\langle \alpha _{\mathrm{Vv}},\alpha _{\mathrm{Hh}}\rangle \\ & & +\langle \alpha _{\mathrm{Hh}},\alpha _{\mathrm{Vv}}\rangle +\langle \alpha _{\mathrm{Vh}},\alpha _{\mathrm{Hv}}\rangle +\langle \alpha _{\mathrm{Hv}},\alpha _{\mathrm{Vh}}\rangle \bigr )\,. {}\end{array}$$

(5.224)

If we set $\omega _{\mathrm{i}} = 0$ and $\omega _{\mathrm{f}} =\pi /2$ in Eq. (5.224), we obtain the relation

$$\displaystyle{ \langle \alpha _{\mathrm{Vh}},\alpha _{\mathrm{Vh}}\rangle =\langle \alpha _{\mathrm{Hv}},\alpha _{\mathrm{Hv}}\rangle \,. }$$

(5.225)

If $\omega _{\mathrm{i}'}$ and $\omega _{\mathrm{f}'}$ are certain values of $\omega _{\mathrm{i}}$ and $\omega _{\mathrm{f}}$ for which the last term on the right-hand side of Eq. (5.224) does not vanish, this term may be expressed as a linear combination of $\langle \alpha _{\mathrm{Vv}},\alpha _{\mathrm{Vv}}\rangle$, $\langle \alpha _{\mathrm{Hv}},\alpha _{\mathrm{Hv}}\rangle$ ($=\langle \alpha _{\mathrm{Vh}},\alpha _{\mathrm{Vh}}\rangle$), $\langle \alpha _{\mathrm{Hh}},\alpha _{\mathrm{Hh}}\rangle$, and $\langle \alpha _{\mathrm{f}'\mathrm{i}'},\alpha _{\mathrm{f}'\mathrm{i}'}\rangle$. Therefore, it turns out that $\langle \alpha _{\mathrm{fi}},\alpha _{\mathrm{fi}}\rangle$ for arbitrary $\omega _{\mathrm{i}}$ and $\omega _{\mathrm{f}}$ may be expressed as a linear combination of $\langle \alpha _{\mathrm{Vv}},\alpha _{\mathrm{Vv}}\rangle$, $\langle \alpha _{\mathrm{Hv}},\alpha _{\mathrm{Hv}}\rangle$, $\langle \alpha _{\mathrm{Hh}},\alpha _{\mathrm{Hh}}\rangle$, and $\langle \alpha _{\mathrm{f}'\mathrm{i}'},\alpha _{\mathrm{f}'\mathrm{i}'}\rangle$. Thus $F_{\mathrm{fi}}$ may be expressed as a linear combination of $F_{\mathrm{Vv}}$, $F_{\mathrm{Hv}}$ ($= F_{\mathrm{Vh}}$), $F_{\mathrm{Hh}}$, and $F_{\mathrm{f}'\mathrm{i}'}$. If we choose as the fourth component $F_{\mathrm{f}'\mathrm{i}'} = F_{\mathrm{Qq}}$ with $\omega _{\mathrm{i}'} =\omega _{\mathrm{f}'} =\pi /4$, we obtain Eq. (5.91).