Square-Gradient Model for Inhomogeneous Systems: From Simple Fluids to Microemulsions, Polymer Blends and Electronic Structure

Abstract

The square-gradient approximation for describing the thermodynamic and structural properties of inhomogeneous systems was first proposed many years ago by van der Waals in his celebrated work on capillarity. Because it entails no a priori knowledge on inter-particle interactions, the mathematical procedure is generically applicable to a wide variety of seemingly unrelated phenomena, regardless of whether the system contains classical or quantum particles. This chapter presents a few common concepts underlying different incarnations of the square-gradient model for simple fluids, microemulsions, polymer blends as well as electronic systems. Pedagogical examples are given to illustrate its applications to describing the physical properties of various inhomogeneous systems including surface tensions, x-ray scattering from microemulsions, and spinodal decomposition in polymer blends. By emphasizing similar ideas used in different subfields of statistical mechanics, the tutorial material may help better understand connections among similar theoretical methods established in different contexts.

Appendices

Appendices

Density-Gradient Expansion

Consider an inhomogeneous system with one-particle number density $\rho$(r). The intrinsic Helmholtz energy can be formally expressed relative to that of a uniform system with density $\rho _{0}$ by a functional Taylor expansion with respect to the local density deviation $_{ }$ $\Delta \rho (\mathbf{r})=\rho (\mathbf{r})-\rho _0$:

$$\begin{aligned} \begin{aligned} F[\rho (\mathbf{r})]&=F_0 +\int {\left. {\frac{\delta F}{\delta \rho (\mathbf{r})}} \right| _0 \Delta \rho \text {(}{} \mathbf{r}\text {)d}{} \mathbf{r}}\\&\quad +\frac{1}{2}\int \int {\left. {\frac{\delta ^{2}F}{\delta \rho \text {(}{} \mathbf{r}\text {)}\delta \rho \text {(}\mathbf{r}^{\prime }\text {)}}} \right| _0 \Delta \rho \text {(}\mathbf{r}\text {)}\Delta \rho \text {(}{} \mathbf{r}^{\prime }\text {)d}\mathbf{r}\text {d}{} \mathbf{r}^{\prime }} +\cdots \end{aligned} \end{aligned}$$

(90)

where $F_{0}$ is the intrinsic Helmholtz energy of the uniform system at the same temperature, and subscript 0 denotes the uniform reference system. According to Eq. (90), the intrinsic free energy F[$\rho$(r)] is fully specified by a set of functions

$$\begin{aligned} K(\mathbf{r}_1 ,\mathbf{r}_2 ,\cdots \mathbf{r}_n )=\delta ^{n}F/\prod _{i=1}^n {\delta \rho (\mathbf{r}_i )} \end{aligned}$$

(91)

where $n=1,2,\ldots .$

The one-body density profile satisfies the variational condition, i.e., it minimizes the grand potential $\Omega$

$$\begin{aligned} \frac{\delta \Omega }{\delta \rho (\mathbf{r})}=0 \end{aligned}$$

(92)

The grand potential relates to the intrinsic Helmholtz energy by the Legendre transformation:

$$\begin{aligned} \Omega [\rho (\mathbf{r})]=F[\rho (\mathbf{r})]+\int {\rho (\mathbf{r})u(\mathbf{r})d\mathbf{r}} \end{aligned}$$

(93)

where $u(\mathbf{r})\equiv \varphi ^{\text {ext}}(\mathbf{r})-\mu$ corresponds to a one-body potential define by the chemical potential and the external potential $\varphi ^{\text {ext}}(\mathbf{r})$ of the particles, $\mu$ is the chemical potential. Combing Eqs. (92) and (93) leads to

$$\begin{aligned} \frac{\delta F}{\delta \rho (\mathbf{r})}=-u(\mathbf{r}) \end{aligned}$$

(94)

For a uniform system, $\varphi ^{\text {ext}}(\mathbf{r})=0$, Eq. (94) reduces to

$$\begin{aligned} \left. {\frac{\delta F}{\delta \rho (\mathbf{r})}} \right| _0 =\mu . \end{aligned}$$

(95)

The second-order term in the density functional expansion of the intrinsic Helmholtz energy plays a particularly important role in theoretical developments. The vertex function is defined as

$$\begin{aligned} K(\mathbf{r},\mathbf{r}^{\prime })\equiv \frac{\delta ^{2}F}{\delta \rho (\mathbf{r})\delta \rho (\mathbf{r}^{\prime })}=-\frac{\delta u(\mathbf{r})}{\delta \rho (\mathbf{r}^{\prime })} \end{aligned}$$

(96)

It can be shown that the density-density correlation function is related to the functional derivative of the one-body density with respect to the reduced one-body potential

$$\begin{aligned} \chi (\mathbf{r},\mathbf{r}^{\prime })\equiv \left\langle {[\hat{{\rho }}(\mathbf{r})-\rho (\mathbf{r})][\hat{{\rho }}(\mathbf{r}^{\prime })-\rho (\mathbf{r}^{\prime })]} \right\rangle =-\frac{\delta \rho (\mathbf{r})}{\delta \beta u(\mathbf{r}^{\prime })} \end{aligned}$$

(97)

where the instantaneous density of the system

$$\begin{aligned} \hat{{\rho }}(\mathbf{r})\equiv \sum _i {\delta (\mathbf{r}-\mathbf{r}_i )} \end{aligned}$$

(98)

is expressed as a summation of Dirac delta functions, and $\left\langle \cdots \right\rangle$ represents the ensemble average.

From Eqs. (96) and (97), we see that the density-density correlation function corresponds to the inverse functional derivative of the 2nd order coefficient in the functional Taylor expansion of the intrinsic Helmholtz energy

$$\begin{aligned} \int {\beta K(\mathbf{r},\mathbf{r}^{{\prime }{\prime }})\chi (\mathbf{r}^{\prime },\mathbf{r}^{{\prime }{\prime }})d\mathbf{r}^{{\prime }{\prime }}} =\delta (\mathbf{r}-\mathbf{r}^{{\prime }{\prime }}). \end{aligned}$$

(99)

For a uniform system, both $K_0 (\mathbf{r},\mathbf{r}^{\prime })$ and $\chi _0 (\mathbf{r},\mathbf{r}^{\prime })$ depend only on the distance $|\mathbf{r}-\mathbf{r}^{\prime }|$. In that case, we may apply the translational and rotational symmetry for the correlation functions:

$$\begin{aligned} \left\{ {{\begin{array}{l} {K(\mathbf{r},\mathbf{r}^{\prime })=K(0,\mathbf{r}-\mathbf{r}^{\prime })\equiv K_0 (|\mathbf{r}-\mathbf{r}^{\prime }|)} \\ {\chi (\mathbf{r},\mathbf{r}^{\prime })=\chi (0,\mathbf{r}-\mathbf{r}^{\prime })\equiv \chi _0 (|\mathbf{r}-\mathbf{r}^{\prime }|)} \\ \end{array} }} \right. \end{aligned}$$

(100)

Substituting Eq. (100) into (99) gives

$$\begin{aligned} \int {\beta K_0 (|\mathbf{r}-\mathbf{r}^{{\prime }{\prime }}|)\chi _0 (|\mathbf{r}^{\prime }-\mathbf{r}^{{\prime }{\prime }}|)d\mathbf{r}^{{\prime }{\prime }}} =\delta (\mathbf{r}-\mathbf{r}^{{\prime }{\prime }}) \end{aligned}$$

(101)

A 3D Fourier transform of Eq. (101), reveals a simple relationship between $\tilde{K}_0 (q)$ and $\tilde{\chi }_0 (q)$

$$\begin{aligned} \beta \tilde{K}_0 (q)\tilde{\chi }_0 (q)=1 \end{aligned}$$

(102)

where

$$\begin{aligned} \tilde{K}_0 (q)=\tilde{K}_0 (\mathbf{q})\equiv \int {K_0 (r)e^{-i\mathbf{q}\cdot \mathbf{r}}\text {d}{} \mathbf{r}} =\frac{4\pi }{q}\int _0^\infty {r\sin (qr)K_0 (r)dr} \end{aligned}$$

(103)

$$\begin{aligned} \tilde{\chi }_0 (q)=\tilde{\chi }_0 (\mathbf{q})\equiv \int {\chi _0 (r)e^{-i\mathbf{q}\cdot \mathbf{r}}\text {d}{} \mathbf{r}} =\frac{4\pi }{q}\int _0^\infty {r\sin (qr)\chi _0 (r)dr} \end{aligned}$$

(104)

Applying Eq. (97) to a uniform system of average density $\rho _0$, we have

$$\begin{aligned} \begin{array}{c} \chi _0 (|\mathbf{r}-\mathbf{r}^{\prime }|)=\chi (\mathbf{r},\mathbf{r}^{\prime })=\left\langle {\hat{{\rho }}(\mathbf{r})\hat{{\rho }}(\mathbf{r}^{\prime })} \right\rangle -\rho _0^2 \\ =\left\langle {\hat{{\rho }}(\mathbf{r})\hat{{\rho }}(\mathbf{r}^{\prime })} \right\rangle _{\mathbf{r}\ne \mathbf{r}^{\prime }} +\left\langle {\hat{{\rho }}(\mathbf{r})\hat{{\rho }}(\mathbf{r}^{\prime })} \right\rangle _{\mathbf{r}=\mathbf{r}^{\prime }} -\rho _0^2 \\ =\rho ^{(2)}(\mathbf{r},\mathbf{r}^{\prime })+\rho _0 \left\langle {\hat{{\rho }}(\mathbf{r}^{\prime })} \right\rangle _{\mathbf{r}=\mathbf{r}^{\prime }} -\rho _0^2 \\ =\rho _0^2 h_0 (|\mathbf{r}-\mathbf{r}^{\prime }|)+\rho _0 \delta (\mathbf{r}-\mathbf{r}^{\prime }) \\ \end{array} \end{aligned}$$

(105)

where $\rho ^{(2)}(\mathbf{r},\mathbf{r}^{\prime })$ is the two-body density function, and

$$\begin{aligned} h_0 (|\mathbf{r}-\mathbf{r}^{\prime }|)\equiv \left. {\frac{\rho ^{(2)}(\mathbf{r},\mathbf{r}^{\prime })}{\rho (\mathbf{r})\rho (\mathbf{r}^{\prime })}} \right| _0 -1 \end{aligned}$$

(106)

denotes the total correlation function. In the Fourier space, Eq. (105) becomes

$$\begin{aligned} \tilde{\chi }_0 (q)=\rho _0^2 \tilde{h}_0 (q)+\rho _0 \end{aligned}$$

(107)

With the help of Eqs. (95), (102) and (107), we can evaluate the intrinsic free energy functional F[$\rho$(r)] up to the quadratic term. Apparently, the density expansion is applicable not only to the intrinsic Helmholtz energy but also to other quantities with a similar mathematic form, for example, the excess free energy and the exchange-correlation energy.

The Ornstein-Zernike (OZ) Equation

Recalling that the vertex function is inversely related to the density–density correlation function $\chi (\mathbf{r}_1 ,\mathbf{r}_2 )$, which is also related to the total correlation function $h(\mathbf{r}_1 ,\mathbf{r}_2 )$

$$\begin{aligned} \chi (\mathbf{r}_1 ,\mathbf{r}_2 )=\rho (\mathbf{r}_1 )\rho (\mathbf{r}_2 )h(\mathbf{r}_1 ,\mathbf{r}_2 )+\rho (\mathbf{r}_1 )\delta (\mathbf{r}_1 -\mathbf{r}_2 ). \end{aligned}$$

(108)

For classical systems, the Helmholtz energy of a non-interacting system is exactly known.

$$\begin{aligned} F^{ID}=k_\text {B} T\int {\rho (\mathbf{r})\{\ln [\rho (\mathbf{r})\Lambda ^{3}]-1\}d\mathbf{r}} . \end{aligned}$$

(109)

Subsequently, we may express the vertex function in terms of the direct correlation function (DCF)

$$\begin{aligned} \beta K(\mathbf{r},\mathbf{r}^{\prime })=\frac{\delta ^{2}\beta F}{\delta \rho \text {(}{} \mathbf{r}\text {)}\delta \rho \text {(}{} \mathbf{r}^{\prime }\text {)}}=\frac{\delta (\mathbf{r}-\mathbf{r}^{\prime })}{\rho (\mathbf{r})}-c(\mathbf{r},\mathbf{r}^{\prime }) \end{aligned}$$

(110)

where $c(\mathbf{r},\mathbf{r}^{\prime })$ corresponds to the second-order functional derivatives of the excess Helmholtz energy $F^{\text {ex}}\equiv F-F^{ID}$

$$\begin{aligned} c(\mathbf{r},\mathbf{r}^{\prime })\equiv -\frac{\delta ^{2}\beta F^{\text {ex}}}{\delta \rho (\mathbf{r})\delta \rho (\mathbf{r}^{\prime })} . \end{aligned}$$

(111)

Because

$$\begin{aligned} \int d \mathbf{r}_2 \chi (\mathbf{r}_1 ,\mathbf{r}_2 )\beta K(\mathbf{r}_3 ,\mathbf{r}_2 )=\delta (\mathbf{r}_1 -\mathbf{r}_3 ), \end{aligned}$$

(112)

substituting Eqs. (110) and (108) into (111) leads to the Ornstein-Zernike (OZ) equation

$$\begin{aligned} h(\mathbf{r}_1 ,\mathbf{r}_2 )=c(\mathbf{r}_1 ,\mathbf{r}_2 )+\int {\rho (\mathbf{r}_3 )h(\mathbf{r}_1 ,\mathbf{r}_3 )c(\mathbf{r}_2 ,\mathbf{r}_3 )d\mathbf{r}_3 } \end{aligned}$$

(113)

For uniform systems, the OZ equation can be simplified as

$$\begin{aligned} h(r)=c(r)+\rho _0 \int {h(|\mathbf{r}_1 -\mathbf{r}_3 |)c(|\mathbf{r}_2 -\mathbf{r}_3 |)d\mathbf{r}_3 } \end{aligned}$$

(114)

or in the Fourier space

$$\begin{aligned}{}[1+\rho _0 h_0 (q)][1-\rho _0 c_0 (q)]=1 . \end{aligned}$$

(115)

Corrections to the Local Density Approximation (LDA)

Local density approximation (LDA) assumes that the free energy density of an inhomogeneous system is the same as that of a uniform system at the local density. According to LDA, the intrinsic Helmholtz energy functional is given by

$$\begin{aligned} F^{\text {LDA}}[\rho (\mathbf{r})]=\int {f_0 [\rho (\mathbf{r})]d\mathbf{r}} \end{aligned}$$

(116)

where $f_0 =F_0 /V$ corresponds to the intrinsic free energy density (per volume) of a uniform system. LDA ignores the spatial correlation effect.

Because LDA assumes $f_0$ as a function of $\rho (\mathbf{r})$, we may express it as a regular Taylor expansion with respect to that of a uniform system

$$\begin{aligned} f_0 [\rho (\mathbf{r})]=f_0 (\rho _0 )+\frac{\partial f_0 }{\partial \rho _0 }\Delta \rho (\mathbf{r})+\frac{1}{2}\frac{\partial ^{2}f_0 }{\partial \rho _0^2 }[\Delta \rho (\mathbf{r})]^{2}+\cdots \end{aligned}$$

(117)

Substituting Eq. (117) into (116), we have:

$$\begin{aligned} F^{\text {LDA}}[\rho (\mathbf{r})]=F_0 +\int {\frac{\partial f_0 }{\partial \rho _0 }\Delta \rho (\mathbf{r})\text {d}{} \mathbf{r}} +\frac{1}{2}\int {\frac{\partial ^{2}f_0 }{\partial \rho _0^2 }[\Delta \rho (\mathbf{r})]^{2}\text {d}{} \mathbf{r}} +\cdots \end{aligned}$$

(118)

Comparing Eq. (118) with the functional expansion form, i.e., Eq. (90), we have:

$$\begin{aligned} \begin{aligned} F[\rho (\mathbf{r})]&=F^{\text {LDA}}[\rho (\mathbf{r})] \\&\quad +\frac{1}{2}\int \int {[K_0 (|\mathbf{r}-\mathbf{r}^{\prime }|)-\left( {\frac{\partial \mu }{\partial \rho _0 }} \right) _T \delta (\mathbf{r}-\mathbf{r}^{\prime })]\Delta \rho (\mathbf{r})\Delta \rho (\mathbf{r}^{\prime })\text {d}{} \mathbf{r}\text {d}{} \mathbf{r}^{\prime }} +\cdots \\ \end{aligned} \end{aligned}$$

(119)

In writing the above equation, we have used the thermodynamic relation

$$\begin{aligned} \mu =(\partial f_0 /\partial \rho _0 )_T \end{aligned}$$

(120)

and the mathematic identity

$$\begin{aligned} \int {\frac{\partial ^{2}f_0 }{\partial \rho _0^2 }[\Delta \rho (\mathbf{r})]^{2}\text {d}{} \mathbf{r}} =\int {\left( {\frac{\partial \mu }{\partial \rho _0 }} \right) _T \Delta \rho (\mathbf{r})\Delta \rho (\mathbf{r}^{\prime })\delta (\mathbf{r}-\mathbf{r}^{\prime })\text {d}{} \mathbf{r}\text {d}{} \mathbf{r}^{\prime }} \end{aligned}$$

(121)

In Eq. (119), the terms after $F$ $^{LDA}$ can be regarded as spatial correlation effects neglected by LDA.

Now let $F_{2}$ represent the second term on right side of Eq. (119). Using the Fourier transform, we can express $F$ $_{2}$ as

$$\begin{aligned} F_2 =\frac{1}{2(2\pi )^{3}}\int {[\tilde{K}_0 (q)-\left( {\frac{\partial \mu }{\partial \rho _0 }} \right) _T ][\Delta \tilde{\rho }(\mathbf{q})]^{2}\text {d}{} \mathbf{q}} \end{aligned}$$

(122)

According to Eqs. (102) and (107), we have

$$\begin{aligned} \beta \tilde{K}_0 (q)=\frac{1}{\rho _0^2 \tilde{h}_0 (q)+\rho _0 } \end{aligned}$$

(123)

In addition, the compressibility equation gives

$$\begin{aligned} \left( {\frac{\partial \rho _0 }{\partial \beta \mu }} \right) _T =\rho _0 +\rho _0^2 \int {h_0 (r)d\mathbf{r}} =\rho _0 +\rho _0^2 \tilde{h}_0 (q=0)=\frac{1}{\beta \tilde{K}_0 (0)} . \end{aligned}$$

(124)

Accordingly, Eq. (122) can be rewritten in a more compact form:

$$\begin{aligned} F_2 =\frac{1}{16\pi ^{3}}\int {[\tilde{K}_0 (q)-\tilde{K}_0 (0)][\Delta \tilde{\rho }(\mathbf{q})]^{2}\text {d}{} \mathbf{q}} \end{aligned}$$

(125)

With Eq. (125), we formulate the additional correlation term beyond LDA. Similar to the functional expansion, such procedure can be extended to other quantities.

To connect Eq. (125) with the square-gradient expansion, we recall that

$$\begin{aligned} \tilde{K}_0 (q)=\tilde{K}_0 (\mathbf{q})\equiv \int {K_0 (r)e^{-i\mathbf{q}\cdot \mathbf{r}}\text {d}{} \mathbf{r}} =\int {\frac{\sin (qr)}{qr}K_0 (r)\text {d}{} \mathbf{r}} \end{aligned}$$

(126)

Using the Taylor series

$$\begin{aligned} \frac{\sin (qr)}{qr}=1-\frac{(qr)^{2}}{3!}+\cdots \end{aligned}$$

(127)

we have

$$\begin{aligned} \begin{array}{l} \tilde{K}_0 (q)=\int {\left[ {1-\frac{(qr)^{2}}{3!}} \right] K_0 (r)\text {d}{} \mathbf{r}} \\ \quad \quad \;=\tilde{K}_0 (0)-\frac{q^{2}}{3!}\int {r^{2}K_0 (r)\text {d}{} \mathbf{r}} \\ \end{array} \end{aligned}$$

(128)

Substituting Eq. (128) into (127) gives

$$\begin{aligned} F_2 =-\frac{\kappa }{16\pi ^{3}}\int {q^{2}[\Delta \tilde{\rho }(\mathbf{q})]^{2}\text {d}{} \mathbf{q}} \end{aligned}$$

(129)

where $\kappa$ is the influence parameter defined as

$$\begin{aligned} \kappa \equiv -\frac{1}{3!}\int {r^{2}K_0 (r)\text {d}{} \mathbf{r}} =\frac{1}{3}\mathop {\lim }\limits _{q\rightarrow 0} \;[K_0 (q)-K_0 (0)]/q^{2} \end{aligned}$$

(130)

Note

$$\begin{aligned} \int {\nabla \rho (\mathbf{r})e^{i\mathbf{q}\cdot \mathbf{r}}d\mathbf{r}} =-i\mathbf{q}\Delta \tilde{\rho }(\mathbf{q}) \end{aligned}$$

(131)

and

$$\begin{aligned} \frac{1}{(2\pi )^{3}}\int {q^{2}[\Delta \tilde{\rho }(\mathbf{q})]^{2}\text {d}{} \mathbf{q}} =\int {d\mathbf{r}_1 \int {d\mathbf{r}_2 \nabla \rho (\mathbf{r}_1 ) } } \nabla \rho (\mathbf{r}_2 )\delta (\mathbf{r}_1 -\mathbf{r}_2 ) \end{aligned}$$

(132)

we arrive the square-gradient correction to the LDA

$$\begin{aligned} F_2 =\frac{\kappa }{2}\int {\text {d}{} \mathbf{r}} [\nabla \rho (\mathbf{r})]^{2} . \end{aligned}$$

(133)

In some applications, we use the static structure factor $\tilde{S}_0 (q)=\tilde{\chi }_0 (q)/\rho _0$ instead of the vertex function. In that case,

$$\begin{aligned} \tilde{K}_0 (q)=[\beta \rho _0 \tilde{S}_0 (q)]^{-1} . \end{aligned}$$

(134)

thus the influence parameter is given by

$$\begin{aligned} \kappa =-\frac{1}{3!}\int {r^{2}K_0 (r)\text {d}{} \mathbf{r}} =\frac{1}{3\beta \rho _0 }\mathop {\lim }\limits _{q\rightarrow 0} \;[\tilde{S}_0^{-1} (q)-\tilde{S}_0^{-1} (0)]/q^{2} . \end{aligned}$$

(135)

Intra-Chain Correlation Function of a Gaussian Chain

In a polymer blend A and B, the intra-chain correlation for polymer A as an ideal chain is given by

$$\begin{aligned} \chi _{0,AA} (|\mathbf{r}-\mathbf{r}^{\prime }|)=\frac{1}{v_0^2 }<\delta \hat{{\phi }}_A (\mathbf{r})\delta \hat{{\phi }}_A (\mathbf{r}^{\prime })>=\frac{\phi _{0,A}}{v_0^{2}}P_\text {A} (|\mathbf{r}-\mathbf{r}^{\prime }|) \end{aligned}$$

(136)

where $P$ $_{A}$ represents the probability to find a segment at position $\mathbf{r}$ given that another segment from the same polymer chain is located at $\mathbf{r}^{\prime }$. A similar expression can be written for polymer B.

For a non-interacting polymer, $P$ $_{A}$ corresponds to a Gaussian average of all segment pairs separated by distance r

$$\begin{aligned} P_\text {A} (r)=\frac{1}{N_\text {A} V}\sum _{i\ne j}^{N_\text {A} } {\int {d\mathbf{r}_i \int {d\mathbf{r}_j p_{ij} (\mathbf{r})} } \delta [\mathbf{r}-(\mathbf{r}_i -\mathbf{r}_j )]} \end{aligned}$$

(137)

where r $_{i}$ and r $_{j}$ represent the position of segment i and j from the same polymer chain, respectively. In Eq. (137), the Gaussian distribution function is given by the random walk model [20]

$$\begin{aligned} p_{ij} (\mathbf{r})=\left( {\frac{3}{2\pi |i-j|l_A^2 }} \right) ^{3/2}\exp \left( {-\frac{3r^{2}}{2|i-j|l_A^2 }} \right) \end{aligned}$$

(138)

where $l_A$ stands for step length or the polymer bond length. Applying the 3-D Fourier transform to both side of Eq. (136) yields

$$\begin{aligned} \tilde{P}_\text {A} (q)=\frac{1}{N_\text {A} }\sum _{i\ne j}^{N_\text {A} } {\exp \left( {-\frac{q^{2}|i-j|l_A^2 }{6}} \right) } \end{aligned}$$

(139)

For a long polymer chain, $N_\text {A} \gg 1$, the double summations in Eq. (139) can be replaced by integrations

$$\begin{aligned} \begin{array}{l} \tilde{P}_\text {A} (q)=\frac{1}{N_\text {A} }\int _0^{N_\text {A} } {\text {d}x\int _0^{N_\text {A} } {\text {d}y\exp \left( {-\frac{q^{2}|x-y|l_A^2 }{6}} \right) } } \\ \quad \quad \quad \,\, =N_\text {A} D(qR_\text {A} ) \\ \end{array} \end{aligned}$$

(140)

where $R_\text {A} \equiv N_\text {A} l_A^2 /6$ is the radius of gyration for an ideal polymer chain (Gaussian chain), and

$$\begin{aligned} D(x)=\frac{2}{x^{4}}\left( {e^{-x^{2}}+x^{2}-1} \right) \end{aligned}$$

(141)

is known as the Debye function. For small x, $D(x)\approx 1-x^{2}/3$, we can derive from Eq. (136) the intra-chain correlation in the Fourier space

$$\begin{aligned} \tilde{\chi }_{0,AA} (q)=\frac{\phi _{0,\text {A}} }{v_0^2 }\tilde{P}_\text {A} (q)\approx \frac{N_\text {A} \phi _{0,\text {A}} }{v_0^2 }\left( {1-\frac{q^{2}R_\text {A}^2 }{3}} \right) . \end{aligned}$$

(142)