Persistence length by De Gennes: how to make sense of the definitional equation?

Question

I want to understand how De Gennes comes up with the concept of persistence length in microemulsions, and in particular a convenient equation that relates this persistence length to the thickness of the film through an exponential term depending on a bending rigidity (for further details on this, please refer to J. Phys. Chem. 86(13) 1982). The equation reads:

$$\xi_K = a\exp\frac {2\pi K}{kT}$$

$\xi_K$ is the persistence length, $a$ is the thickness of the interface, $K$ is the bending rigidity (in energy units), $k$ is Boltzmann constant and $T$ is temperature. This equation is pivotal to understand phase diagrams of microemulsions. However, the demonstration in the article itself is very dry and it seems to me that there are inconsistencies in dimensional analysis, or typographic errors, and this makes it impossible for me to understand the demonstration.

Commented demonstration for persistence length

The demonstration starts with setting a model system with a reference plane on xy coordinates with a negligible spontaneous curvature and seeks to study the fluctuations of curvature. Then De Gennes defines $\zeta(xy)$ as the distance between the fluctuating interface and the reference plane. De Gennes states a geometrical definition of the curvature and equates it with a mathematical one (if I do this by myself using the parametric equation of a sphere for small curvatures I get a 1/2 factor in front of the differential). He writes:

$$\frac{1}{R} = \frac{\partial^2\zeta}{\partial x^2} + \frac{\partial^2\zeta}{\partial y^2} \equiv \Delta_\bot\zeta (1)$$

Then he states the Helfrich equation when the interfacial tension is zero, which defines the elastic free energy for a bending rigidity $K$ as:

$$f_K = \int\frac{1}{2}K(\Delta_\bot\zeta)^2 = \sum_{q}\frac{1}{2}Kq^4|\zeta_q|^2 (2) $$

Here already I have a problem. The bending rigidity has the dimension of an energy, and $\Delta_\bot\zeta$ has the dimension of inverse length. However the integral does not have integrands such as, eg, $dxdy$. Is this a typo or is this normal? What are the upper and lower limits of integration? Without the integrands the integrals even though it is called a free energy in fact has the dimension of a surface tension.

Then he doesn't define $q$ here even though from context (because he talks about Fourier transforms later on) I presume that this is the magnitude of a wave vector, and therefore he is representing the fluctuations of the planar interface as a superposition of waves. To me $|\zeta_q|^2$ represents the mean square displacement on the interface for a wave with wave vector q. I have a hard time understanding whether $\pi/q$ represents the length of an individual wave or its amplitude.

We then have a definition for $\zeta_q$ which is in fact a complex number due to the use of a 2-dimensional Fourier transform (that's the only way for me to understand the || symbols in the previous equation even though I thought that the overall complex integral would somehow make it a real number):

$$\zeta_q = \int dxdy \zeta(xy) \exp{[i(q_x x + q_y y)]} (3)$$

That very specific equation has puzzled me a lot. I tried to perform that Fourier transform by myself but it seems that there is a missing factor $\int dxdy$ somewhere and I can't figure out where it is. I've seen a complex integral that resolves to 0 when the thing in the exponent is not zero and to 1 otherwise but it would need that normalization factor. Only in this condition I can get the equivalence in eq. 2 right. Furthermore I would expect that $\zeta_q$ has the dimension of a distance but it seems it has the dimension of a volume, which makes it even more puzzling to me! Is it me or is there some kind of a typo there?

Then he defines a normal vector that characterises the local orientation of the surface $\textbf{n}$ as:

$$n_x = -\partial\zeta/\partial x \\\ n_y = -\partial\zeta/\partial y (4)\\\ n_z \approx 1 $$

and he goes on to define a small fluctuation of this 3d vector $\delta\textbf{n}$ as $\delta\textbf{n} = (n_x, n_y)$ which suddenly is 2d vector.

It appears that he applies the derivative once on eq. 3 leading to

$$|\delta\textbf{n}|^2 = q^2|z_q|^2 (5)$$

Here I think that $z_q$ is a typo for $\zeta_q$ because otherwise it doesn't make any sense as this $z$ variable appears nowhere else in the article. It looks like the exponential is derived to get this $q^2$ factor but I'm not even sure of this. Also if I follow correctly this fluctuation variable has no unit because it's a squared inverse distance times a squared distance. Except if actually $\zeta_q$ is a volume, in this case it is distance to the fourth power which makes it completely physically nonsensical to me. Therefore I have to assume that this is a distance and then that there is a typo in eq. 3.

He then applies the equipartition theorem to give $kT/2$ energy for each degree of freedom, ie for each "mode" $q$ (he calls these modes here). well, I do obtain the same thing than he does, i. e., for a given q, I get what he writes:

$$\langle |\delta\textbf{n}_q|^2\rangle = \frac{kT}{Kq^2} (6) $$

But even though I get this equation by combining eqs. 2 and 5 and applying equipartition theorem, it doesn't make any sense to me. Indeed the equation has dimension squared distance due to the $q^2$. $kT$ and $K$ are both energies and therefore cancel each other.

He then defines a variable called "angular correlation" which really gives me headaches. This is the most obscure part of the demonstration to me:

$$\theta^2(r) = \langle |\delta\textbf{n}(0) - \delta\textbf{n}(r)|^2\rangle = \sum_{q} 2[1-cos(\textbf{q} \cdot \textbf{r})]\langle |\delta\textbf{n}_q|^2\rangle (7)$$

$\textbf{r}$ is defined as a position on the surface, ie as far as I understand as a $x,y$ pair. Then I must assume that $r$ is the magnitude of this vector. Except for that, simply put, I don't understand anything here. I don't understand how an angle is defined as a difference between two vectors because angles are scalar, not vectors. I don't understand what it means to make the difference $\delta\textbf{n}(0) - \delta\textbf{n}(r)$ because I don't see anything special to the point at $r = 0$. I don't understand how out of the blue there is this $[1-cos(\textbf{q} \cdot \textbf{r})]$ factor that appears. I am completely lost here. And De Gennes puts the nail in my coffin by adding this equality just below:

$$=\frac{kT}{\pi K}\int_{0}^{1/a}[1-J_0(qr)]\frac{dq}{q} (8)$$

$a$ is defined as a molecular size, ie $1/a$ is a very big, but not infinite, inverse length.

What? What is a Bessel function doing here? How does the sum suddenly transforms into an integral? why is $\textbf{q} \cdot \textbf{r}$ suddenly becoming simply $qr$? That doesn't make any sense to me.

On the basis that $1-J_0(qr)$ is 1 for $x >> 1$ and 0 for $x << 1$ (ok, that is mathematics, I can accept that), he obtains the following equation from above:

$$\langle \theta^2 \rangle = \frac{kT}{\pi K} \ln \frac{r}{a} (9)$$

where in the absence of specification on his part I must assume that $\langle \theta^2 \rangle$ is just another way to write $\theta^2(r)$. Otherwise it's even more nonsensical to me. I have to note that this very equation isn't transparent to me. the integral of $\frac{dq}{q}$ when evaluated in the lower limit gives $ln(1/a)$ and in the lower limit it gives $ln(0)$, i. e. it diverges. Even though the factor $1-J_0(qr)$ gives 0 at the lower limit, how can he achieve an analytic formula for this?

Then he applies a few approximations valid for small $\theta$ (or small fluctuations) which seem less of a problem to me even though I don't understand how he changes the application of average brackets $\langle\rangle$:

$$\langle \cos \theta \rangle \approx \langle 1 - \frac{\theta^2}{2} \rangle \approx \exp \left( - \frac{\langle \theta^2 \rangle}{2} \right) = \left(\frac{a}{r}\right)^{kT/(2\pi K)} (10)$$

assuming $\cos \theta = 1/e$ as the cross over value upon which $\theta$ angles are no longer considered small, a persistence length $\xi_K$ (taking the place of the variable $r$), describing the length of the "approximately planar" sections of the interface, is defined:

$$\xi_K = a\exp\frac {2\pi K}{kT} (11)$$

which is the equation I seek to understand. I have no problem with the last step, it sounds reasonable and clear to me.

Trying to grasp analogous equations in liquid crystals

As reading the original article did not help me a lot to understand the equation, I sought some books from De Gennes where he might expand upon these concepts. The only one that seems somewhat close is "The Physics of Liquid Crystals". In this book, at some point, he makes some Fourier transforms and ends up writing these equations:

crystal: $$\langle u^2(r)\rangle = \frac{kT}{8\pi^3C}\int_{L^{-1}}^{q_c}\frac{dq^3}{q^2}\approx\frac{kT}{C}\frac{q_c}{2\pi^2}$$ columnar phase: $$\langle u^2(r)\rangle = \frac{kT}{8\pi^3C}\int_{L^{-1}}^{q_c}\frac{dq^3}{q_\bot^2+\lambda^2q_z^4}\approx\frac{kT}{C}\frac{\sqrt{q_c/\lambda}}{2\pi^2}$$ smectic: $$\langle u^2(r)\rangle = \frac{kT}{8\pi^3C}\int_{L^{-1}}^{q_c}\frac{dq^3}{q_z^2+\lambda^2q_\bot^4}\approx\frac{kT}{C}\frac{\ln q_cL}{4\pi\lambda}$$

As far as I understand it, $\langle u^2(r) \rangle$ is the average squared displacement with respect to a reference plane, $C$ is an elastic constant with dimensions of a pressure (like a modulus), $q$ is a wave vector, with the length of a period being something like $\pi/q$, $\lambda$ is a characteristic length of about 5 angstroms (probably the average molecular diameter when looking at the interface from above), $q_c$ is a critically high wave vector to the point that elastic theory is not valid anymore, (i. e. $\pi/q_c$ reaches molecular dimensions), and $L$ is the overall domain of integration (which could be infinity in the case it does not diverge)

Mathematically, I don't understand these integrals. What does it mean to integrate $d^3q/q^2$? Can someone decompose the elementary steps allowing to first evaluate and then approximate these integrals? The online integral evaluators don't help me with these because of the weird $d^3q$.

Answer 1

We need to make sense of $$\int_{L^{-1}}^{q_c}\frac{dq^3}{q^2}\approx4\pi q_c,\,\int_{L^{-1}}^{q_c}\frac{dq^3}{q_\bot^2+\lambda^2q_z^4}\approx4\pi\sqrt{\frac{q_c}{\lambda}},\,\int_{L^{-1}}^{q_c}\frac{dq^3}{q_z^2+\lambda^2q_\bot^4}\approx\frac{2\pi^2}{\lambda}\ln(q_cL)$$(note the brackets needed to make the last one clear). The symbol $d\vec{q}^3$ (the author didn't include the $\vec{}$), or (probably) more often $d^3\vec{q}$, means $dq_xdq_ydq_z$, i.e. these definite integrals are over a $3$-dimensional $\vec{q}$-space. This is of course the "reciprocal" space of wavevectors. The limits may look one-dimensional, but are constraints on the radius $q:=|\vec{q}|$, i.e. the integration range is the finite volume between two origin-centred spheres.

All three integrals have an integrand of dimension $[q^{-2}]=\mathsf{L}^2$ and integration operator of dimension $[d^3\vec{q}]=\mathsf{L}^{-3}$, so the integrals have dimension $\mathsf{L}^2\mathsf{L}^{-3}=\mathsf{L}^{-1}$. These match the given values because, as you already know,$$[q_c]=[q_\perp]=[q_z]=\mathsf{L}^{-1},\,[\lambda]=[L]=\mathsf{L}.$$

Since in spherical polar coordinates $d^3\vec{q}=q^2dq\sin\theta d\theta d\varphi$, the first integral is$$\int_{L^{-1}}^{q_c}dq\int_0^\pi\sin\theta d\theta\int_0^{2\pi}d\varphi=4\pi(q_c-L^{-1}).$$The approximation $q_cL\gg1$ rewrites this as $4\pi q_c$. We can change the lower limit to $0$ in the first two integrals (indeed, I'll do that next with the second), but such a step would cause the logarithm in the third approximation to diverge.

Since $q_\perp= q\sin\theta,\,q_z=q\cos\theta$, the second integral is$$2\pi\int_0^\pi d\theta\int_0^{q_c}\frac{dq}{\sin^2\theta+q^2\lambda^2\cos^4\theta}=\frac{2\pi}{\lambda}\int_0^\pi\frac{d\theta}{\cos^2\theta}\arctan\left(\frac{q_c\lambda\cos^2\theta}{\sin\theta}\right),$$so the hard part is proving$$\int_0^\pi\frac{d\theta}{\cos^2\theta}\arctan\left(\frac{X\cos^2\theta}{\sin\theta}\right)\approx2\sqrt{X},\,X:=q_c\lambda.$$Similarly, the third integral is equivalent to$$\int_0^\pi\frac{1}{\cos\theta}\left[\arctan\left(q_c\lambda\tan\theta\right)-\arctan\left(L^{-1}\lambda\tan\theta\right)\right]\approx\pi\ln(q_cL).$$

To address your edit:

• $f_k$'s first definition indeed omits an area element; physicists often drop the $dx$ in $\int fdx$, but they shouldn't.
• It looks like powers of $R$ have been forgotten in a few places; for example, if $\zeta_q$ is defined as in (3) so $[\zeta_q]=\mathsf{L}^3$, we need an $R^{-2}$ factor in (2)'s sum, an $R^{-4}$ factor in (5) etc. However, that may not be the correct read of this situation. One subtlety in comparing integrals to sums is that sums don't have an analogous $[dxdy]=\mathsf{L}^2$ factor. Therefore, I suspect $[\zeta q]=\mathsf{L}$, (2) needs an $R^2$ factor etc.
• For (7), $\vec{r}$ denotes the displacement between two points whose angular correlation is sought (so $\vec{0}$ is actually quite special). I can be shown we seek the mean of the square of a vector of length $2\left|\sin\tfrac{\vec{q}\cdot\vec{r}}{2}\right|\left|\delta\vec{n}_q\right|$. (You can understand this with complex exponentials, or a diagram of a subtended chord.) I can't remember how one gets to a $1-J_0$ factor in (8).
• For (9), the approximation gives $\int_0^{1/a}[1-J_0(qr)]\frac{dq}{q}\approx\int_{1/r}^{1/a}\frac{dq}{q}=\ln\frac{r}{a}$.
• For (10), $\left\langle1-\tfrac{\theta^2}{2}\right\rangle=1-\left\langle\tfrac{\theta^2}{2}\right\rangle\approx\exp\left(-\left\langle\tfrac{\theta^2}{2}\right\rangle\right)$.