Elastic Properties of Symmetric Liquid-Liquid Interfaces

TL;DR

This study uses self-consistent field theory to accurately calculate the elastic bending rigidities of symmetric liquid-liquid interfaces, revealing their signs and scaling behavior near criticality and at strong segregation.

Contribution

It provides high-precision, mean-field results for the bending rigidities of symmetric liquid-liquid interfaces using a specific SCF model, clarifying their signs and dependence on system parameters.

Findings

Both moduli are positive near criticality, scaling with interfacial tension.

At strong segregation, Gaussian rigidity becomes negative.

Chain length influences the length scale  significantly at small N.

Abstract

The mean ($\kappa$) and Gaussian ($\bar{\kappa}$) bending rigidities of liquid-liquid interfaces, of importance for shape fluctuations and topology of interfaces, respectively, are not yet established: even their signs are debated. Using the Scheutjens Fleer variant of the self-consistent field theory, we implemented a model for a symmetric L/L interface and obtained high precision (mean field) results in the grand canonical $(\mu, V, T)$-ensemble. We report positive values for both moduli when the system is close to critical where the rigidities show the same scaling behavior as the interfacial tension $\gamma$. At strong segregation, when the interfacial width becomes of the order of the segment size, $\bar{\kappa}$ turns negative. The length scale $\lambda \equiv \sqrt{\kappa/\gamma}$ is of order the segment size for all strengths of interaction; yet the $1/\sqrt{N}$ chain length correction reduces $\lambda$ significantly when the chain length $N$ is small.

Equations

$$\gamma (J,K)=\gamma (0,0)+\frac{\partial \gamma }{\partial J}J+\frac{1}{2}\frac{{\partial }^2\gamma }{\partial J^2}{J}^2+\frac{\partial \gamma }{\partial K}K+\cdots$$

$$F=-\ln Q([u])-\sum _{x,r}{u}_x(r){\phi }_x(r)L(r)+F^{\mathrm{int}}([\phi ])+$$

$$\displaystyle\sum_{r}\alpha(r)\bigg{[}\sum_{x}\varphi_{x}(r)-1\bigg{]}+\Delta_{r;r_{0}}\bigg{[}\varphi_{A}(r)-\varphi_{B}(r)\bigg{]}$$

$$F^{\rm int}=\chi\sum_{r}L(r)\varphi_{A}(r)\bigg{[}\langle\varphi_{B}(r)\rangle-\varphi_{B}^{b}\bigg{]}$$

$$⟨{\phi }_B(r)⟩={\phi }_B(r)+\frac{1}{6}{\nabla }^2{\phi }_B(r)$$

Full text

††preprint:

Out-of-plane fluctuations of liquid-liquid (L/L) interfaces are controlled by the interfacial tension ($\gamma$) for wavelengths larger than the cross-over length $\lambda=\sqrt{\kappa/\gamma}$ and by the bending rigidity ($\kappa$) at shorter wavelengths. This $\lambda$ should be comparable to a molecular length scale Laradji and Mouritsen (2000). Precise prediction of $\lambda$ from a molecular model would significantly advance our understanding on fluctuations in L/L interfaces. However, molecular models that have access to $\kappa$ at sufficient accuracy have not yet been forwarded. More specifically, a key issue here is that molecular theories thus far have failed to establish the sign of $\kappa$.

As $\kappa$ controls the magnitude of the fluctuations (at short length scales), we expect it to be positive ($\kappa>0$). In stark contrast to this, surprisingly few theoretical predictions foresee a positive value Blokhuis and Bedeaux (1992): to date, molecular models typically predict negative values Matsen (1999); Leermakers (2013a); Oversteegen and Blokhuis (2000); Blokhuis et al. (1999). Nevertheless in (mesoscale) simulations Van Giessen and Blokhuis (2002); Müller and Gompper (2002); Müller and Schmid (2005) and in phenomenological models Laradji and Desai (1998) a positive sign for $\kappa$ is often chosen.

Besides $\kappa$, interfaces have a second elastic constant known as the saddle spay modulus or Gaussian bending rigidity $\bar{\kappa}$. The $\bar{\kappa}$ should control the topology of interfaces; a negative value will prevent the formation of saddle shaped interfaces whereas a positive value will promote these. A sign-change (e.g. upon a change of temperature) is easily envisioned. However, existing predictions indicate a strictly positive value Matsen (1999); Leermakers (2013a); Oversteegen and Blokhuis (2000).

Molecular theories give relatively easy access to the accurate values for the interfacial tension Van der Waals (1873, 1979). However, the evaluation of the rigidities has many intricacies. With respect to common practise, we found that a sound estimation of $\kappa$ and $\bar{\kappa}$ from a molecular theory require to overcome two hurdles: (i) to quantify the curvature expenses at a fixed chemical potential ($\mu$) of all molecular species and (ii) to properly account for non-local contributions to the enthalpic interactions.

In this letter, we successfully overcome these theoretical challenges and show that $\kappa$ is strictly positive for L/L interfaces and hence fluctuations from planar state cost free-energy. We observe that $\lambda$ is of the order of the segment size in the limit of strong or weak segregation, yet shows a non-monotonous behavior in transition regime. We discuss the implications of chain length ($N$ - degree of polymerization) on fluctuations of L/L interfaces, in light of results obtained for $\lambda$. Finally, we present and discuss the sign-switch for $\bar{\kappa}$.

Mean field results for a simple symmetric interface between two liquids $A_{N}$ and $B_{N}$, where $N$ is chain length (degree of polymerization), is discussed. The case with $N=1$ correspond to the well-known van der Waals interface Van der Waals (1979); Safran (1994). When $N$ is large we arrive at another well studied interface, namely between two inmiscible polymers Semenov (1993); Matsen (1999). As $\gamma>0$, the system has a tendency to minimize its area. Thermal energy causes the macroscopic interface to fluctuate. The accompanied entropy gain is counteracted by an unfavourable increase in area and a penalty for the interface to (locally) bend away from the planar ground state. Such curved interfaces cannot maintain their tension exactly. Following Helfrich Helfrich (1973) we consider a Taylor series expansion of the tension in terms of the mean ($J=1/R_{1}+1/R_{2}$) and Gaussian ($K=1/R_{1}\times 1/R_{2}$) curvature ($R_{1}$ and $R_{2}$ are two principle radii of curvature):

[TABLE]

The term linear in $J$ is well documented and properly understood Rowlinson and Widom (2013); Mitrinović et al. (2000, 2000); Wu et al. (1993); Ocko et al. (1994); Merkl et al. (1997). Here and below, we will focus on symmetric interfaces for which this linear term vanishes. Defining the mean bending modulus, $\kappa\equiv\frac{\partial^{2}\gamma}{\partial J^{2}}$, and Gaussian bending rigidity, $\bar{\kappa}\equiv\frac{\partial\gamma}{\partial K}$, Eqn. 1 reduces for weakly curved interface to $\gamma(J,K)-\gamma(0,0)=\frac{1}{2}\kappa J^{2}+\bar{\kappa}K$. The grand potential, $\Omega=\int\gamma{\rm d}A$, quantifies the excess free energy of the interface and is the characteristic function that can be used to describe the fluctuations of the interface that take place at specified chemical potentials, $\mu$, (that of the binodal) of the molecular species in the system.

We implement a self-consistent field model using lattice approximations as introduced by Scheutjens and Fleer for polymer adsorption Scheutjens and Fleer (1979, 1980). These authors combined a freely jointed chain model with a Flory-Huggins equation of state. The repulsive interactions between $A$ and $B$ segments is quantified by a Flory-Huggins interaction parameter. When $\chi>\chi^{c}=2/N$ the system features a solubility gap. It turns out that it is important to understand how the SF-SCF formalism deviates from the classical SCF theory that is used to describe microphase segregation (which is also frequently used to model the interface between two polymeric solutions).

In SF-SCF we write a mean field free energy (in dimensionless units) for a molecularly inhomogeneous system Leermakers (2013b); Kik et al. (2010); Fleer et al. (1993); Evers et al. (1990) in terms of volume fraction $\varphi_{x}({\bf r})$ and complementary segment potential $u_{x}({\bf r})$ profiles for segment types $x=A,\ B$ on a grid of lattice sites with characteristic size $b$ equal to segment size. To facilitate proper extremization we add a Lagrange parameters, $\alpha({\bf r})$, in free energy to implement the local incompressibility constraint, $\varphi_{A}(r)+\varphi_{B}(r)=1$, applicable in each coordinate $r$ and a parameter $\Delta_{r;r_{0}}\equiv\delta_{r;r_{0}}\nu$, where $\delta_{r;r_{0}}$ is unity when $r=r_{0}$ and zero otherwise and $\nu$ is the Lagrange parameter coupled to the requirement that at the interface location $r=r_{0}$ the density of both components match Matsen (1999):

[TABLE]

In the mean field approach one can decompose the partition function $Q=\Pi_{i}q_{i}^{n_{i}}/n_{i}!$ where $i=1,\ 2$ refers to $A_{N}$ and $B_{N}$ respectively. The molecular partition function $q_{i}$ contains the statistical weight of all possible and allowed conformations of molecule $i$ (see below). $n_{i}$ is the number of molecules of type $i$ in the system. $L(r)$ gives the number of lattice sites at the lattice coordinate $r$. For planar system $L(r)=1$ is a constant (all quantities are per unit area), in cylindrical coordinates $L(r)\propto r$ (and quantities are expressed per unit length of the cylinder), while in spherical coordinates $L(r)\propto r^{2}$. The interaction free energy is a function of the densities:

[TABLE]

Here $\varphi_{B}^{b}$ refers to the bulk volume fraction of B (of one of the bulk phases). Importantly, the angular brackets are needed to account for the contact energy in a system with gradients in density. Similar as in the Cahn Hilliard theory Safran (1994) we write

[TABLE]

where the $\nabla^{2}$ is easily mapped on the lattice as explained extensively in earlier literature Scheutjens and Fleer (1979); Leermakers (2013b). SCF solutions now involve optimizing the free energy ($F$) with respect to its variables, respectively segment potentials, volume fractions. When $\partial F/\partial\varphi_{A}({r})=0$, we find that the potentials must obey $u_{A}({r})=\alpha(r)+\Delta_{r;r_{0}}+\chi(\langle\varphi_{B}(r)\rangle-\varphi_{B}^{b})$ (and similarly for $u_{B}(r)$). Setting $\partial F/\partial u_{x}(r)=0$ shows the way to evaluate the densities: $\varphi_{x}(r)=-\partial\ln Q/\partial u_{x}(r)$. The molecular partition functions are found from the endpoint distribution $q_{i}=\sum_{r}G_{i}(r,N|1)$. The end-point distributions are recursively found from $G_{i}(r)=\exp-u_{i}(r)$ by the propagator $G_{i}(r,s|1)=G_{i}(r)\langle G_{i}(r,s-1|1)\rangle$, where the angular brackets have the same meaning as in Eqn.4. The segment densities are found by the composition law, which for homopolymers read: $\varphi_{i}(r)=C_{i}G_{i}(r,s|1)G_{i}(r,N-s-1|1)/G_{i}(r)$. As the position of the interface is already controlled by a Lagrange parameter $\Delta_{r;r_{0}}$, we no longer need to specify a fixed amount of one of the components (as is needed in a canonical ensemble), but we can normalize the densities with $C_{i}=\varphi_{i}^{b}/N_{i}$ where $\varphi_{i}^{b}$ is specified by the binodal: A proper binodal value is a (relevant) root of the Flory-Huggins Eqn. $\ln\frac{\varphi}{1-\varphi}+\chi N(1-2\varphi)=0$.

Numerical solutions, which routinely were obtained with an accuracy of 9 significant digits, that optimize the free energy functional and obey to all constraints, have the property that the potentials both determine and follow from the volume fractions profiles and vice versa and are said to be self-consistent. For such solution one can compute the grand potential by $\Omega=\sum_{r}L(r)\omega(r)$ wherein the grand potential density at coordinate $r$ is given by \omega(r)=-\sum_{i}(\varphi_{i}({r})-\varphi_{i}^{b})/N_{i}-\alpha({r})-\chi(\varphi_{A}({r})\big{[}\varphi_{B}({r})+\frac{1}{6}\nabla^{2}\varphi_{B}({r})\big{]}-\varphi_{A}^{b}\varphi_{B}^{b}). The planar interface has a tension $\gamma_{p}=\sum_{z}\omega(z)$, where $z$ is the coordinate in the planar system. This planar interface serves as the ground state or reference state needed to estimate the grand potential increase of the curved interfaces.

SCF solutions are routinely created for planer (p) cylindrical (c) and spherical (s) coordinates. As the position of the interface is exactly known and specified by $\Delta_{r;r_{0}}$ to be at coordinate $r=r_{0}$ we obtain the interfacial tensions in all cases unambiguously. In spherical geometry we have $R\equiv R_{1}=R_{2}=r_{0}-1/2$, while in cylindrical geometry $R\equiv R_{1}=r_{0}-1/2$ and $R_{2}=\infty$ and $\gamma_{s}=\Omega_{s}/(4\pi R^{2})$ and $\gamma_{c}=\Omega_{c}/(2\pi R)$ for spherical and cylindrical geometries, respectively. Here $\Omega_{s}$ is the grand potential when the interface is curved in a spherical fashion, and $\Omega_{c}$ is the grand potential per unit length of the interface when curved in a cylindrical fashion. Here we have implemented that a lattice site at coordinate $r$ is a distance $r-1/2$ away from the center of the coordinate system. In all calculations we make sure that the numerical value for $r_{0}$ significantly exceeds the width of the interface. Next we compute $\gamma_{c}-\gamma_{p}$ as well as $\gamma_{s}-\gamma_{p}$ that use the Helfrich equation 1 to extract with high accuracy both $\kappa$ (from cylindrical geometry) and $2\kappa+\bar{\kappa}$ (from spherical geometry). The combination of these results leads to both $\kappa$ and $\bar{\kappa}$. Note that in all calculations, $\mu$ for the molecular species were set to the value at the appropriate binodal. Invariably, we find a positive value for the mean bending modulus whereas the sign of $\bar{\kappa}$ is (as expected) not fixed.

Results for $N=1$ (the van der Waals interfaceVan der Waals (1873, 1979)) are presented in SI, here we focus on the captivating results for $N>1$ and understand that $N=1$ is a limiting case.

In Fig. 1(a), we present the interfacial tension and in Fig. 1(b) the bending rigidities, both as a function of $\Delta\chi$, where $\Delta\chi\equiv\chi-\chi^{c}$ ($\chi^{c}=2/N$: bulk critical point), for three values of the chain lengths $N=2,\ 20$ and $200$. The corresponding results for the density difference and the interfacial width are presented sup . The results for the interfacial tension are in principle well known Safran (1994); Helfand and Tagami (1971); Helfand (1975); Matsen (1999); Semenov (1993, 1994). As long as the interface is wide compared to the coil size we find $\gamma\propto N(\Delta\chi)^{3/2}$ and in the other limit, where the interfacial width is small compared to the coil size, we have $\gamma\propto(\Delta\chi)^{1/2}$. (The 3/2 exponent is the mean field prediction, known to be subject to changes, the 1/2 exponent is expected to be accurate.) As in this regime $\chi$ exceeds $\chi^{c}$ a lot, the result is similar to the known result that $\gamma\propto\sqrt{\chi}$. The latter result/regime is referred to as the strong segregation limit and the former regime will be referred to as the weak segregation limit.

Interestingly the results for both bending rigidities [cf. Fig. 1(b)] follow the results for the interfacial tension qualitatively. In the weak segregation the $3/2$ scaling is found, while for the strong segregation the $1/2$ scaling is recovered. It is important to mention that the Gaussian bending modulus deviates from the latter power-law dependence rather abruptly: quite suddenly the Gaussian bending rigidity goes to zero and then changes its sign. We will discuss this behaviour below in more detail. Comparing Figs. 1(a) and 1(b) shows that in the transition regions between weak and strong segregation the tension and rigidities do not exactly copy their dependencies. This has interesting implications as we will show next.

Above we introduced the length scale $\lambda=\sqrt{\kappa/\gamma}$. Combining results form Fig. 1(a), 1(b), we present $\lambda$ as a function of $\Delta\chi$ in semi-logarithmic coordinates for a few systems that differ in $N$ in Fig. 2(a). In this figure it can be seen that $\lambda$ goes through a local maximum ($\lambda^{\rm max}$) in-between weak and strong segregation. Further, $\lambda$ reaches some fixed value, which is approximately 16% larger at weak-, compared to strong segregation. In Fig. 2(b), linear dependence of $\lambda^{\rm max}$ is presented as a function of chain length $N$. Computer simulations that are aimed to find the bending rigidity from the height fluctuations of the interfaces Müller et al. (1995); Van Giessen and Blokhuis (2002) may benefit from relatively large $\lambda$-values and preferably should be executed for large chains, because $\lambda^{\rm max}$ grows linearly with $N$ as $\lambda^{\rm max}=0.02N+6.54$ .

It is of interest to consider the ratio $\bar{\kappa}/\kappa$. Obviously, when $\bar{\kappa}$ is switching its sign, we cannot have a fixed ratio between the rigidities, but slightly outside the transition regions, that is both at weak and at strong segregation, the ratio is remarkably constant. This is illustrated in Fig. 3(a) where for a selected number of values for $\Delta\chi$-values this ratio is plotted as a function of the chain length $N$. For low values of $N$ and low values of $\Delta\chi$ we have the typical weak segregation result. It could have been concluded from Fig. 1(b), the ratio goes to a constant of approximately $1/2$. In the high $N$ high $\Delta\chi$ value limit, that is, the strong segregation limit, where the Gaussian bending rigidity is negative, the ratio is approximately $-3/2$. Implicit in this prediction is that also in the regime where $\bar{\kappa}<0$, the (negative) Gaussian bending rigidity follows a scaling law: $-\bar{\kappa}\propto(\Delta\chi)^{1/2}$.

Recently, we have shown for microemulsion systems that the Gaussian bending rigidity is positive for systems near the (bulk) critical point and negative otherwise Varadharajan and Leermakers (2018). From Figs. 1(b) and 3(a) it can be seen that for liquid/liquid interfaces the same phenomenology applies and it is of interest to quantify this sign switch. It is mentioned in SI sup that the scaling for the interfacial width $W\propto(\Delta\chi)^{-1/2}$ applies both at weak and strong segregation regimes. The appropriate prefactor depends on $1+1/N$ in weak segregation and on $1+1/\sqrt{N}$ in strong segregation regime sup ; Mocan et al. (2018). In Fig. 3(b) we present a diagram of states in coordinates $\sqrt{\Delta\chi}$ vs $1/\sqrt{N}$. The dashed line ($(\Delta\chi)^{1/2}=0.2+0.5/\sqrt{N}$) separates the parameter combination with positive $\bar{\kappa}$ values from the negative ones. The line in Fig. 3(b) is functionally consistent with the prefactor for the interfacial width scaling in the strong segregation limit and therefore we speculate that the interfacial width controls the sign switch. Apparently, the Gaussian bending rigidity changes sign when the width of the interface is approximately 3 to 4 times the segment size. This thus happens at relatively weak segregation for short chains but in the strong segregation regime for long chains.

Physically, the implication of a positive $\kappa$ is that fluctuations of the L/L interface away from the planar state do cost (free)energy. In the light of existing literature, this expected result is remarkable for several reasons.

(i) The way the interface is pinned, using a Lagrange parameter coupled to the equal density of the two liquid component, was first used by Matsen Matsen (1999). He implemented this method to find the bending rigidities using the classical self-consistent field machinery. His SCF approach has many similarities with the (current) SF-SCF approach. Yet he reported strictly negative values for $\kappa$ and positive values for $\bar{\kappa}$. The only relevant difference between our SF-SCF approach and the classical SCF approach of Matsen rests in the fact that for the interactions we have implemented the Cahn-Hilliard gradient terms, and Matsen did not. In the SI we show that when in SF-SCF these gradient terms are neglected, that is, when we implement (cf Eqn 4) $\langle\varphi(r)\rangle\to\varphi(r)$, we do reproduce all results of Matsen to a high accuracy. This proves the importance of the non-local interaction contributions to determine the rigidities. In the absence of these non-local interactions, neither the sign nor the scaling dependencies are apparently properly predicted.

(ii) A number of years ago Blokhuis has shown that a big effect on how bending of the interface is implemented Blokhuis (2009) must be expected. He identified the so-called equilibrium bending mode where $\mu$ controls the curvature. In this case, the bending of the interface is accompanied by the development of a Laplace pressure inside the ‘droplet’ phase. $\gamma$ is then typically computed at the so-called surface of tension (SOT). The position of the interface is taken to be at the SOT, even though other choices can be implemented. When the tension evaluated at this SOT is used in the Helfrich equation one can evaluate $\kappa$ and $\bar{\kappa}$ (again using the combination of cylindrical and spherical geometries). As confirmed by SF-SCF calculations Leermakers et al. (2004); Oversteegen and Blokhuis (2000), in this case $\kappa$ is negative and $\bar{\kappa}$ is strictly positive. Also for equilibrium bending one finds scaling behaviour for both moduli when the system is close to the bulk critical point. However in this case the coefficient of $1/2$ is found. Blokhuis also analyzed the bending of the interface at fixed $\mu$ (binodal values) by controlling the position of the interface by some (local) external field. Interestingly, in this case he recovered the 3/2 scaling law, similarly as presented above for the weak segregation Blokhuis et al. (2008). However, still the value of $\kappa$ was negative. Interestingly, quantitative values for $\kappa$ did depend on the choice that was implemented to define the interface position. Blokhuis could not exclude that there might be some choices for this that could turn $\kappa$ positive. Hence the current results that shows positive values for $\kappa$ and a 3/2 scaling coefficient (in the weak segregation limit) indeed is the anticipated result.

Perhaps the more interesting prediction is the sign switch of $\bar{\kappa}$. In surfactant systems such sign switch has been found earlier Varadharajan and Leermakers (2018) and is expected because it correlates with the rich phase behavior for these systems that include cubic phases and sponge phases. For the liquid/liquid interface the sign switch of $\bar{\kappa}$ is unknown. It will be of more than average interest to find experiments for which this sign switch is important. This is not trivial because we know that the prime interest of an interface is to reduce its area under the influence of a finite tension. However, when the interfaces are strongly perturbed, one might find that drops may pinch off. Arguably this is easier when $\bar{\kappa}<0$, hence at strong segregation systems, and suppressed otherwise. This reasoning may explain why for some liquids one can manipulate the splashing by an external pressure Xu et al. (2005); Yarin (2006). Such effects may find applications in various industrial process Tekin et al. (2008); Duez et al. (2007). Our results may also have implications in emulsion droplet formation as the ease by which drops form might be manipulable by the sign and size of $\bar{\kappa}$.

We have proved that the fluctuations from L/L interface away from the planar interface indeed cost free energy. We have shown that the cross-over length has a non-monotonous behavior in the transition regime between weak and strong segregation. Besides this, $\lambda$ is essentially constant (of the order of a few segments lengths) and does not vary much with chain length and/or distance to a critical point. Moreover, a sign-switch of $\bar{\kappa}$ is now established. As interfaces are omnipresent, it is difficult to overestimate the many implications of our phenomenal results which may include complex phenomena such as droplet nucleation from a supersaturated solution, emulsion formation and wetting phenomena to mention a few.

This work is part of an Industrial Partnership Programme, ‘Shell/NWO Computational Sciences for Energy Research (CSER-16)’, of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO). Project number: 15CSER26.