The Interface in Demixed Colloid–Polymer Dispersions

Abstract

In Chaps. 3 and 4, the focus was on theory and experiments related to the phase behaviour of mixtures containing colloidal spheres and nonadsorbing polymers. As we have seen, when the polymer coils are sufficiently large relative to the colloidal spheres, a colloidal gas–liquid (fluid–fluid) phase separation may occur. The two phases that appear differ in composition. One phase is a dilute colloidal fluid (a colloidal ‘gas’) dispersed in a concentrated polymer solution. This phase coexists with a concentrated colloidal fluid (a colloidal ‘liquid’) dispersed in a dilute polymer solution.

In Chaps. 3 and 4, the focus was on theory and experiments related to the phase behaviour of mixtures containing colloidal spheres and nonadsorbing polymers. As we have seen, when the polymer coils are sufficiently large relative to the colloidal spheres, a colloidal gas–liquid (fluid–fluid) phase separation may occur. The two phases that appear differ in composition. One phase is a dilute colloidal fluid (a colloidal ‘gas’) dispersed in a concentrated polymer solution. This phase coexists with a concentrated colloidal fluid (a colloidal ‘liquid’) dispersed in a dilute polymer solution. Besides the phase behaviour, the properties of the interface between such coexisting phases have gained interest [1,2,3,4,5,6,7,8,9,10]. The interface can be characterised by a number of quantities, such as the interfacial tension and the interfacial thickness. Perrin’s atom–colloid analogy  suggests similarities with the molecular gas–liquid interface. However, as we will see, there are also differences driven by the vastly different length scales of the systems.

As discussed in Sect. 1.3.2, we expect the interfacial tension in demixed colloid–polymer systems to be ultra low. Its magnitude can be estimated from [11]

$$\begin{aligned} {} \gamma \approx \frac{kT}{d^2}, {}\end{aligned}$$

(5.1)

where d is the particle diameter. For simple molecular systems, where d is less than roughly a nanometer, this yields values for the surface tension $\gamma$ of about 10–100 mN/m [12], which agrees well with experimental results. For the colloidal domain, interfacial tensions are predicted to be orders of magnitude smaller; for instance, for particles of $d = 100$ nm an interfacial tension of about 0.4 $\upmu$N/m is predicted. As we shall see, this is indeed the order of magnitude of the tension of the colloidal gas–liquid interface that is found in experiments and theory.

In Sect. 5.1, we focus on experiments that have been conducted to measure the interfacial tension. Subsequently, a theoretical approach is presented in Sect. 5.2 that enables one to predict the interfacial tension. This is accompanied by a quantification of the thickness of the interface, which will be compared with experiments. In Sect. 5.3, we show that the ultra-low tension gives rise to an interfacial roughness due to thermal fluctuations, which can be observed visually. The implications of these aspects for the hydrodynamics of droplet coalescence are also discussed.

5.1 Interfacial Tension Measurements

Although various approaches to determine interfacial tensions exist, such as the Wilhelmy plate, the Du Noüy ring or the pendant drop techniques [13], these common methods generally do not allow (accurate) measurement of the ultra-low tensions occurring in demixed colloid–polymer mixtures. Here, we highlight two methods that have been used successfully in the past to measure these interfacial tensions: the spinning drop and the interfacial profile methods.

5.1.1 The Spinning Drop Method

In the spinning drop method [14, 15], a droplet of the phase with the lowest density (usually the colloidal gas phase) is suspended in the phase with the highest density (usually the colloidal liquid phase) in a tube (Fig. 5.1). When spinning this tube around its axis, the elongation of this droplet induced by centrifugal forces is balanced by interfacial forces. As shown below, it is possible to quantify the interfacial tension using this force balance from an analysis of the droplet deformation.

Fig. 5.1

Parts reproduced with permission from Ref. [16]. Copyright 2008 Springer

Measurement of the tension of the colloidal gas–liquid interface using spinning drop measurements. A droplet of the colloidal gas phase with the indicated dimensions is dispersed in the colloidal liquid phase. The system is composed of stearyl silica spheres in cyclohexane ($d = 26$ nm) with poly(dimethylsiloxane) ($q \approx 1.1$).

When the length L of the droplet is significantly larger than its diameter D, the droplet geometry approaches the so-called Vonnegut limit [14]. In this case, the balance between centrifugal and interfacial forces can be quantified as follows: consider two points (C) and (D) located at a distance D/2 from the axis of rotation, as depicted in Fig. 5.1. For mechanical equilibrium, the pressure at these two points must be equal: $P_\mathrm {(C)} = P_\mathrm {(D)}$. These pressures $P_\mathrm {(C)}$ and $P_\mathrm {(D)}$ are related to the pressures at points (A) and (B) that lie on the axis of rotation. The pressure at (C) is the pressure at (A) plus a contribution by the centrifugal pressure, $P_\mathrm {(C)} = P_\mathrm {(A)} + \frac{1}{8}\rho _\textrm{L} \omega ^2 D^2$, where $\omega$ is the angular velocity and $\rho _\textrm{L}$ is the mass density of the liquid phase. Similarly, the pressure at point (D) is the pressure at (B) plus the centrifugal pressure minus the Laplace pressure due to the cylindrically shaped interface at this position; as such, $P_\mathrm {(D)} = P_\mathrm {(B)} + \frac{1}{8}\rho _\textrm{G} \omega ^2 D^2 - \frac{2\gamma }{D}$, where $\rho _\textrm{G}$ is the mass density of the colloidal gas phase. In turn, the pressure at point (B) is the pressure at point (A) plus the Laplace pressure due to the curvature of the spherically shaped end-cap of the droplet, which has a radius a, so that $P_\mathrm {(B)} = P_\mathrm {(A)} + \frac{2\gamma }{a}$. It can be shown that, in the Vonnegut limit, $a = \frac{1}{3}D$ (see, for instance, Chap. 1 of Ref. [13]). Combining these ingredients, we find [14, 15]

$$\begin{aligned} {} \gamma &= \frac{\varDelta \rho \omega ^2 D^3}{32}, {}\end{aligned}$$

(5.2)

where the mass density difference $\varDelta \rho \equiv \rho _\textrm{L} - \rho _\textrm{G}$. This Vonnegut limit is accurate when $L \gtrsim 4D$ [14]; a more general expression by Princen et al. [15] is available that also works for less elongated droplets, for which a is a function of the droplet aspect ratio L/D.

Exercise 5.1. Derive Eq. (5.2) based on the conditions for mechanical equilibrium, centrifugal pressure and Laplace pressure, as outlined in the text.

The spinning drop method was applied to measure the tension of colloidal gas–liquid interfaces by Vliegenthart and Lekkerkerker [17] in 1997. Two years later, more systematic experiments were carried out by de Hoog and Lekkerkerker [18] and Chen et al. [3]. De Hoog and Lekkerkerker studied a system composed of sterically stabilised silica spheres (with diameter $d = 26\mathrm {\ nm} \pm 19 \%$) dispersed in cyclohexane and mixed with poly(dimethylsiloxane) (PDMS, $R_\textrm{g} =$ 14 nm), and found values between approximately 3.0 and 4.5 $\upmu$N/m depending on the colloid and polymer volume fractions (Table 5.1). Chen et al. studied a similar system with slightly smaller silica spheres ($d = 20.2\mathrm {\ nm} \pm 8 \%$) mixed with PDMS in cyclohexane, especially focused on a wide range of polymer concentrations. They found values ranging from about 0.6 to 17 $\upmu$N/m, close to and far from the critical point,, respectively. These values are remarkably close to those predicted by the crude estimate of Eq. (5.1) (6 and 10 $\upmu$N/m, respectively), provided the system is sufficiently far from the critical point.

Exercise 5.2. Based on Eq. (5.2), show that interfacial tensions of the order of micronewtons per metre are indeed experimentally measurable for droplets with a diameter of the order of hundreds of micrometres and $\omega$ of the order of 100 rad/s.

Table 5.1 Measurements of the colloidal gas–liquid interfacial tension by de Hoog and Lekkerkerker [18] for a system comprising sterically stabilised silica spheres with $d = 26\mathrm {\ nm} \pm 19\%$ dispersed in cyclohexane with poly(dimethylsiloxane) ($R_\textrm{g} =$ 14 nm; $q = 1.1$). Note that $\phi$ and $\phi _\textrm{p}$ indicate the overall colloid volume fraction and relative polymer concentration $c_\textrm{p}/c_\textrm{p}^*$, respectively

5.1.2 The Meniscus Method

Another method used to measure ultra-low interfacial tensions is analysing the shape of the interface, known as a meniscus, near a vertical surface. Far from this surface, the interface is macroscopically horizontal as gravity dominates. Near the wall, however, interfacial effects start to dominate as the interface must meet the wall under a certain contact angle, set by the interfacial tension between the two bulk phases and between each of the bulk phases and the surface. As a result, the interface deforms over a certain length scale (Fig. 5.2). This length scale is known as the capillary length and quantifies the balance of gravitational and interfacial forces; it can be found by analysing the shape of the meniscus. Aarts et al. [19] presented the first such measurements for a colloidal gas–liquid interface in the early 2000s. The capillary length is defined as

$$\begin{aligned} {} \ell _\textrm{c} &\equiv \sqrt{\frac{\gamma }{ \varDelta \rho g}}, {}\end{aligned}$$

(5.3)

where $\varDelta \rho$ again is the mass density difference between the two phases and g is the gravitational acceleration.

The balance between the gravitational pressure on one hand and the Laplace pressure due to the deformation of the interface on the other hand can be mathematically expressed as

$$\begin{aligned} {} -\varDelta \rho g z(y) = \frac{\gamma }{R_\textrm{c}(y)}, {}\end{aligned}$$

(5.4)

where z(y) is the height of the interface at a vertical distance y from the wall and $R_\textrm{c}(y)$ is the radius of curvature at that position. Here, we define z(y) such that it is zero far from the wall ($y\rightarrow \infty$), where the interface is flat. The minus sign on the left-hand side signifies that the gravitational contribution to pressure decreases with increasing height.

From geometrical arguments, it can be shown that the radius of curvature can be expressed as

$$\begin{aligned} {} \frac{1}{R_\textrm{c}(y)} = \frac{-z''(y)}{[1+z'(y)^2]^{3/2}}, {}\end{aligned}$$

(5.5)

where $z'(y)$ and $z''(y)$ represent the first and second derivatives of z(y) with respect to y. The minus sign on the right-hand side is due to the convention that an interface that is curved towards the dense bottom phase has a positive radius of curvature. Combining with Eqs. (5.3) and (5.4) gives

$$\begin{aligned} {} z(y) = \ell _\textrm{c}^2 \frac{z''(y)}{[1+z'(y)^2]^{3/2}}, {}\end{aligned}$$

(5.6)

which can be integrated once to yield

$$\begin{aligned} {} \frac{1}{2}z^2 = - \ell _\textrm{c}^2 \left( \frac{1}{\sqrt{1+z'^2}}-1\right) , {}\end{aligned}$$

(5.7)

where the term $-1$ between the parentheses is an integration constant that follows from the boundary condition that $z'(y\rightarrow \infty ) = 0$. In turn, this can be rearranged to give

$$\begin{aligned} {} \frac{\textrm{d} z}{\textrm{d} y} = - \sqrt{\left( \frac{1}{1-\frac{z^2}{2\ell _\textrm{c}^2}}\right) ^2-1}. {}\end{aligned}$$

(5.8)

Unfortunately, this differential equation needs to be solved numerically for z(y). However, its inverse,

$$\begin{aligned} {} \frac{\textrm{d} y}{\textrm{d} z} = - \left[ \left( \frac{1}{1-\frac{z^2}{2\ell _\textrm{c}^2}}\right) ^2-1\right] ^{-1/2}, {}\end{aligned}$$

(5.9)

can be solved analytically. The result is [20]:

$$\begin{aligned} {} \frac{y(z)}{\ell _\textrm{c}} &= {\text {acosh}}\mathopen {}\left( \frac{2\ell _\textrm{c}}{z}\right) - {\text {acosh}}\mathopen {}\left( \frac{2\ell _\textrm{c}}{h}\right) - \sqrt{4-\frac{z^2}{\ell _\textrm{c}^2}} + \sqrt{4-\frac{h^2}{\ell _\textrm{c}^2}}. {}\end{aligned}$$

(5.10)

Here, $h = z(y = 0)$ is the contact height (i.e., the elevation of the interface at the wall), which is given by $h^2 = 2 \ell _\textrm{c}^2(1-\sin \theta )$, where $\theta$ is the contact angle.

The physical interpretation of the capillary length $\ell _\textrm{c}$ becomes clearer if one considers the shape of the profile somewhat away from the wall, where the slope $|z'(y)| \ll 1$. Then Eq. (5.5) may be approximated as $1/R_\textrm{c}(y) = -z''(y)$, and Eq. (5.6) takes the form $z''(y) = z/\ell _\textrm{c}^2$. As a result, z(y) can now explicitly be approximated as

$$\begin{aligned} {} z(y) = z_0 \exp (-y/\ell _\textrm{c}), {}\end{aligned}$$

(5.11)

where $z_0$ is a numerical pre-factor. Thus, it follows that the capillary length can be seen as a transverse (lateral) correlation length.

Exercise 5.3. Make a plot of Eqs. (5.10) and (5.11) for a contact angle of $\theta = 0^{\circ }$. What is the $y/\ell _\textrm{c}$-range of validity of Eq. (5.11)? Estimate the value of $z_0$. How is the validity of the approximation affected if the contact angle is increased towards $90^{\circ }$?

As we have seen, $\gamma$ is quite small for colloidal systems; as such, $\ell _\textrm{c}$ is typically a few orders of magnitude smaller than for molecular systems. The consequence is that perturbations of the interface decay over a short distance, which makes the interface seem flat even close to a wall, as shown in Fig. 5.2a.

Fig. 5.2

Reproduced with permission from Ref. [19]. Copyright 2003 IOP Publishing, Ltd. [19]

Measurement of the tension of a colloidal gas–liquid interface through analysis of the meniscus shape near a vertical wall. a Macroscopic observation of a sample with a width of 1 cm; b micrograph of an area of 0.706 mm $\times$ 0.528 mm; c interfacial profiles fitted to Eq. (5.10).

However, on a scale of tens of micrometres the meniscus is still present (Fig. 5.2b). Using microscopic observations on an appropriate model system, Aarts et al. [19] were able to apply this approach for the first time on a demixed colloid–polymer system. The system was the same as that described for the spinning drop method, composed of sterically stabilised silica spheres ($d = 26\mathrm {\ nm} \pm 19\%$) dispersed in cyclohexane, to which PDMS with $R_\textrm{g} = 14 \mathrm {\ nm}$ was added. Their fits of the experimentally observed menisci are shown in Fig. 5.2c and the results are summarised in Table 5.2, showing tensions that are approximately the same as those obtained via the spinning drop method.

Table 5.2 Measurements of the colloidal gas–liquid interfacial tension by Aarts et al. [19] for a system comprising sterically stabilised silica spheres with $d = 26\mathrm {\ nm} \pm 19\%$ dispersed in cyclohexane with poly(dimethylsiloxane) ($R_\textrm{g} =$ 14 nm; $q = 1.1$). The state points I–IV correspond to those in Fig. 5.2

5.2 Prediction of Interfacial Properties Using Free Volume Theory

In this section, we will focus on predicting properties of colloidal gas–liquid interfaces based on (G)FVT; in particular, we will treat the interfacial tension and interfacial width. It should be stressed that other useful approaches, such as density functional theory (DFT) [21,22,23], provide additional detailed microscopic information. Given the scope of the previous chapters, here we focus mainly on FVT and its extensions. We first discuss the interfacial tension and, subsequently, the interfacial width. Predictions of both these quantities are compared to experiments.

5.2.1 Interfacial Tension

It is possible to extend free volume theory, which usually focuses on the bulk phase behaviour, in such a way that it predicts some properties of interfaces too. This can be done by supplementing the bulk free energy by an additional interfacial term that is proportional to gradients in concentration. This approach was pioneered by van der Waals for molecular systems as early as 1893 [24]. In 2004, Aarts et al. [25] applied the van der Waals approach to FVT for describing the colloidal gas–liquid interface. Here, we will not go into significant theoretical detail; instead, the focus will be on the general principle. For a translation of van der Waals’ original work, see Ref. [26]; for a modern account of van der Waals’ theory, see Chap. 3 of the book of Rowlinson and Widom [11].

The van der Waals approach starts by defining a function $w(\phi )$, which expresses the variation of the free energy per unit volume in the direction perpendicular to the interface, as a function of the local colloid volume fraction $\phi (z)$. It is convenient to work with a dimensionless version of this function, $\widetilde{w}(\phi ) = w v_0/(kT)$, which is defined as [11, 24]

$$\begin{aligned} {} \widetilde{w}(\phi )=\widetilde{\varOmega }(\phi )-\phi \widetilde{\mu }_\text {coex}+\widetilde{\varPi }_\text {coex}, {}\end{aligned}$$

(5.12)

with $\widetilde{\varOmega } = \varOmega v_0/(kTV)$ a dimensionless version of the semi-grand potential common in (G)FVT. Additionally, $\widetilde{\mu }_\text {coex} = \mu /kT$ is the colloid chemical potential and $\widetilde{\varPi }_\text {coex} = \varPi v_0/(kT)$ is the osmotic pressure under the conditions of the two-phase coexistence; these should be computed beforehand using the phase diagram. This definition ensures that $\widetilde{w}(\phi ) = 0$ if $\phi$ equals either of the two coexisting colloid volume fractions, and that it is larger than zero otherwise.

Van der Waals showed that, following this approach, the interfacial tension may be calculated immediately: prior knowledge of the volume fraction profiles $\phi (z)$ is not needed. The interfacial tension can be calculated from $\widetilde{w}(\phi )$ according to

$$\begin{aligned} {} \gamma = 2 \left( \frac{6}{\pi }\right) ^{3/2} \frac{kT}{d^2} \int _{\phi ^\text {g}}^{\phi ^\text {l}}\sqrt{\widetilde{m}\widetilde{w}(\phi ) }\text {d} \phi , {}\end{aligned}$$

(5.13)

where $\phi ^\text {g}$ and $\phi ^\text {l}$ are the coexisting colloid volume fractions of the colloidal gas and liquid phases, respectively. Notice that the prefactor contains the term $kT / d^2$, which was the simple estimation for the magnitude of the interfacial tension introduced in Eq. (5.1). The integral contains a coefficient $\widetilde{m}$, which dictates how much the system (dis)likes deviations from the bulk composition and which is related to the interactions between the colloids. (For simplicity, we use the dimensionless $\widetilde{m}$, defined as $\widetilde{m} = m/(kT d^5)$ where m is the dimensionful quantity.) Effectively, $\widetilde{m}$ quantifies how particles are affected by being inside a concentration gradient at an interface, having fewer interactions on one side than on the other. Mathematically, $\widetilde{m}$ is related to the direct correlation function $c(\widetilde{r})$, which quantifies the direct interactions between two colloidal spheres in the Ornstein–Zernike equation (i.e., excluding the influence of a third colloid), and is related to the probability of finding two particles at a certain distance. The parameter $\widetilde{m}$ is the second moment of this direct correlation function:

$$\begin{aligned} {} \widetilde{m} = \frac{\pi }{3}\int _0^\infty c(\widetilde{r})\widetilde{r}^4\text {d}\widetilde{r}. {}\end{aligned}$$

(5.14)

Here, $\widetilde{r}\equiv r/d$ is the normalised centre-to-centre distance between two colloidal spheres.

In the case of long-ranged interactions, which are typical for colloidal gas–liquid coexistence, the mean-spherical approximation can be accurate [25, 27, 28], which entails $c(\widetilde{r}) = - W(\widetilde{r})/kT$ for $\widetilde{r}\ge 1$, where the depletion pair potential $W(\widetilde{r})$ reads

$$\begin{aligned} {} \frac{W(\widetilde{r})}{kT} = -\int _0^{\phi _\text {d}^\text {R}}\text {d}{\phi _\text {d}^\text {R}}'\left( \frac{\partial \widetilde{\varPi }_\text {d}^\text {R}}{\partial {\phi _\text {d}^\text {R}}'}\right) \frac{V_\text {overlap}}{v_\text {d}}, {}\end{aligned}$$

(5.15)

where ${V_\text {overlap}}/{v_\text {d}}$ is the overlap volume of depletion zones on two spheres normalised by the volume of the depletants. It is given by

$$\begin{aligned} {} \frac{V_\text {overlap}}{v_\text {d}} &= q^{-3} (1+q^*-\widetilde{r})^2 \left( 1+q^*+\frac{\widetilde{r}}{2}\right) , {}\end{aligned}$$

(5.16)

for $1\le \widetilde{r} \le 1+q^*$ (it is zero otherwise). The parameter $q^* = 2\delta _\textrm{s}/d$ denotes the actual size of the depletion zones under the conditions at hand, whereas $q = 2R_\textrm{g} / d$ has its usual meaning.

Fig. 5.3

Tension of the colloidal gas–liquid interface as calculated using (G)FVT for penetrable hard spheres, polymers in $\varTheta$-solvent and polymers in good solvent ($q = 1.1$), compared to measurements on a system of stearyl silica spheres in cyclohexane ($d = 26$ nm) mixed with poly(dimethylsiloxane) (points) by Aarts et al. [19]

With these ingredients, the interfacial tension may be computed for the various (G)FVT flavours that have been discussed in the preceding chapters, i.e., the penetrable hard sphere (PHS) model and the $\varTheta$- and good-solvent conditions for the interacting polymers model. Figure 5.3 shows the interfacial tension as a function of the colloidal liquid–gas volume fraction difference $\varDelta \phi = \phi ^\textrm{l}-\phi ^\textrm{g}$. The quantity $\varDelta \phi$ is a useful measure to quantify how strongly a system is phase separated, i.e., how far it is from the critical point, because it often can be measured quite accurately experimentally and is by definition zero at the critical point.

The calculations are compared to results by Aarts et al. [19], again for a system comprising sterically stabilised silica and PDMS in cyclohexane (a good solvent for PDMS). The results show that the precise polymer model being used strongly affects the magnitude of the calculated interfacial tension; when an appropriate model is used, a good description of experimental results appears to be possible. Using both a van der Waals approach and a DFT approach, Moncho-Jordá et al. [5, 29] have also noted a similar decrease of the interfacial tension when taking into account polymer interactions in their theoretical predictions.

Fig. 5.4

Some data was replotted from Ref. [30]

Phase diagrams (a) and (b) are calculated using (G)FVT for penetrable hard spheres, polymers in a $\varTheta$ solvent and polymers in a good solvent ($q = 1.1$). c Phase diagram of sterically stabilised silica spheres in cyclohexane ($d = 29.4 \pm 2.2$ nm) mixed with poly(dimethylsiloxane) ($R_\textrm{g} \approx 16.4$ nm, 117 kDa), shown for qualitative comparison only.

The (too) high interfacial tensions predicted by the PHS model can be understood from the phase diagrams, which are shown in Fig. 5.4. These reveal that, compared to the other two models, the PHS description has a triple point that is located at substantially higher polymer reservoir volume fractions. In the interacting polymer description, the depletion thickness is reduced upon increasing polymer concentration (see Sect. ), which disfavours a stable colloidal liquid phase at elevated polymer concentrations. This effect is not part of the PHS description, thus yielding a much higher triple point, especially at relatively large q.

The location of the triple point for PHSs at high $\phi _\textrm{p}^\textrm{R}$ implies that large values of $\varDelta \phi$ also occur at high polymer reservoir concentrations. In turn, this means that strong attractions are operational between the colloidal spheres and that the factor $\widetilde{m}$ is also large. Therefore, the PHS description predicts significantly larger interfacial tensions than the interacting polymer models. This once again stresses the importance of being careful in selecting an appropriate polymer model to describe experimental results on colloid–polymer mixtures.

5.2.2 Interfacial Density Profiles

Through the van der Waals approach interfacial density profiles (or volume fraction profiles) may also be computed. This enables a subsequent quantification of the width of such interfaces. In the spirit of the van der Waals approach, the colloid density profile $\phi (z)$ may be found by (numerically) solving the following differential equation:

$$\begin{aligned} {} -\widetilde{w}[\phi (\widetilde{z})] + \frac{6}{\pi }\widetilde{m}\left( \frac{\text {d}\phi (\widetilde{z})}{\text {d}\widetilde{z}}\right) ^2 = 0, {}\end{aligned}$$

(5.17)

or, in dimensionful quantities,

$$\begin{aligned} {} -w[n(z)] + m \left( \frac{\text {d}n(z)}{\text {d}z}\right) ^2 = 0. {}\end{aligned}$$

(5.18)

Here, $\widetilde{z} = z/d$ denotes the distance perpendicular to the interface. Note the appearance of the factor $\widetilde{m}$, which links the strength of the depletion interaction to the broadness of the interface. It follows that small values of $\widetilde{m}$ favour sharper interfaces, and larger values favour broader interfaces. It should be stressed that it is the combination of $\widetilde{w}$ and $\widetilde{m}$ that determines the actual width of the interface: without the contribution of $\widetilde{m}$, profiles would be infinitely sharp; without the contribution of $\widetilde{w}$, profiles would be infinitely wide.

Fig. 5.5

Colloid density profiles of the gas–liquid interface for $q = 1.1$ for penetrable hard spheres (left), polymers in $\varTheta$-solvent, (middle) and polymers in good solvent (right). The dashed curves are fits to the error function Eq. (5.20) to extract a measure for the interfacial width

In Fig. 5.5, we show the colloid density profiles (solid curves) as calculated for $q = 1.1$, where $z < 0$ denotes the colloidal liquid phase and $z > 0$ denotes the colloidal gas phase. Note that the definition of $z = 0$ is arbitrary; here, we have opted to define $z = 0$ such that $\phi (z=0)$ is exactly the average of $\phi ^\textrm{g}$ and $\phi ^\textrm{l}$. These density profiles have been computed for the PHS model (left panel), and the $\varTheta$ (middle) and good (right) solvent scenarios for the interacting polymers description. In each case, the polymer reservoir volume fraction was chosen to obtain (approximately) the indicated colloid density differences $\varDelta \phi$. Qualitatively, it is already evident that, far from the critical point, the density profiles have a width comparable to the colloid diameter except for the PHS model: the large value of $\widetilde{m}$ starts to dominate over $\widetilde{w}$ far from the critical point and disfavours strong gradients. Closer to the critical point the width significantly increases. Also, it appears that for the same $\varDelta \phi$ value there are differences between the ensuing density profiles for the various depletant models.

Before turning towards a more quantitative interpretation of the density profiles, it is worth mentioning that, in principle, one could use the colloid density profiles to also compute the polymer density profiles by employing an appropriate expression for the free volume fraction $\alpha$. It is to be expected that the profiles are qualitatively similar. We are, however, not aware of any experimental data on such polymer density profiles and, therefore, we do not go into further detail on this aspect.

Direct measurements of the full colloid density profiles $\phi (z)$ have also proven challenging. Experiments have been conducted that do not directly assess the full shape of $\phi (z)$ but still shed light on the broadness [30, 31]. The question is, however, how to assign a (unique) thickness to profiles such as those in Fig. 5.5. The answer is, in fact, that there are various ways to do this and, indeed, different experimental approaches may probe different measures for the interfacial width. Therefore, let us first focus on a number of ways in which the width of $\phi (z)$ can be quantified mathematically.

A measure for the interfacial broadness is the so-called 10–90% width, here denoted as $h_\text {10--90}$, which can be found by finding the positions $z_{10}$ and $z_{90}$ in the profiles for which the condition holds that $\phi (z_{10}) = \phi ^\textrm{g} + 0.1 \varDelta \phi$ and $\phi (z_{90}) = \phi ^\textrm{g} + 0.9 \varDelta \phi$, where $h_\text {10--90} = |z_{10} - z_{90}|$. The advantage of this approach is that it does not make assumptions about the shape of $\phi (z)$ [32].

One may also fit the profile $\phi (z)$ to a given function and quantify the width through the resulting fitting parameters. For instance, close to the critical point in mean-field theories, $\phi (z)$ is exactly described by a $\tanh$ profile [33], which may be expressed as

$$\begin{aligned} {} \phi (z) = \phi ^\textrm{g}+\frac{1}{2}\varDelta \phi \left[ 1+\tanh \left( \frac{z}{h_{\tanh }}\right) \right] , {}\end{aligned}$$

(5.19)

which has a width given by $h_{\tanh }$ [34]. It can be shown that the 10–90% width of such a profile is about $h_\text {10--90} \approx 2.2 h_{\tanh }$.

A similar fit function that can be convenient is the cumulative normal distribution, given by

$$\begin{aligned} {} \phi (z) = \phi ^\textrm{g}+\frac{1}{2}\varDelta \phi \left[ 1+{\text {erf}}\left( \frac{1}{\sqrt{2}}\frac{z}{h_{{\text {erf}}}}\right) \right] , {}\end{aligned}$$

(5.20)

where the width $h_{{\text {erf}}}$ is in fact the standard deviation of the distribution. The broadness of this profile can also be expressed in terms of a 10–90% width, $h_\text {10--90} \approx 2.6 h_{{\text {erf}}}$. The $\tanh$ and ${\text {erf}}$ profiles are qualitatively very similar in shape, and their widths are approximately related as $h_{\tanh } \approx 1.2 h_{{\text {erf}}}$. As such, one can see that although the interfacial width can be quantified in various ways, the resulting measures are, in fact, closely related. In Fig. 5.5, the density profiles are fitted to Eq. (5.20) (grey dashed curves); the fit results are being shown in Table 5.3.

Table 5.3 Parameters obtained by fitting Eq. (5.20) to the profiles in Fig. 5.5

We now turn our attention to the few experiments that have been devoted to this topic. De Hoog et al. [31] carried out a pioneering ellipsometric study on a colloidal gas–liquid interface for sterically stabilised silica with $d = 26\mathrm {\ nm} \pm 19\%$ in cyclohexane, mixed with PDMS ($q \approx 1.1$). For a specific sample, which according to the phase diagram in their work has $\varDelta \phi \approx 0.3$, they could obtain a value for the interfacial width. In their analysis, a $\tanh$ profile was assumed, which yielded $h_{\tanh } = 5.2\mathrm {\ nm}$. Such a value for the interfacial width implies $h_{{\text {erf}}}/d \approx 0.17$ (see also Table 5.4), and that the interface is quite a bit sharper than expected, when compared to the ‘good solvent’ scenario in Table 5.3.

More recently, Vis et al. [30] characterised the interfacial structure of a similar system of sterically stabilised silica ($d = 29.4 \pm 2.2\mathrm {\ nm}$) mixed with PDMS ($R_\textrm{g} \approx 16.4\mathrm {\ nm}$) in cyclohexane ($q = 1.1$, good solvent conditions) using X-ray reflectometry. The measured phase diagram for that system is shown in Fig. 5.4 (right panel). In the analysis, it was assumed that the interface followed a cumulative normal distribution, i.e., Eq. (5.20). The results are summarised in Table 5.4.

Table 5.4 Overview of measurements by de Hoog et al. [31] and Vis et al. [30] of the width of colloidal gas–liquid interfaces

Fig. 5.6

Comparison of the interfacial width normalised to a particle diameter as predicted by (G)FVT (curves, [30]) and measured using X-ray reflectometry ($\circ$, [30]) and ellipsometry ($\blacksquare$, [31]) for systems comprising sterically stabilised silica spheres mixed with poly(dimethlysiloxane) in cyclohexane ($q = 1.1$)

A graphical overview of the (G)FVT predictions and the experiments by de Hoog et al. [31] and Vis et al. [30] is given in Fig. 5.6. Overall, the agreement between the latter experiments and the predictions by GFVT (the interacting polymer description) is quite good; the polymer solvency conditions have minimal influence on the theoretically predicted width. In comparison, the experiments by de Hoog et al. [31] seem to find a relatively narrow interface.

In terms of the phase behaviour, we have seen that the PHS model mostly resembles the $\varTheta$-solvent model (Fig. 5.4) in the vicinity of the critical point. For the interfacial width, also the good solvent model gives quite similar results to the other two descriptions. However, at $\varDelta \phi \gtrsim 0.3$, the PHS model starts to display unphysical behaviour in the form of a strongly increasing width. This again has to do with the high polymer concentration of the triple point: the resulting large values of $\widetilde{m}$ yield broader interfaces through Eq. (5.17). Such behaviour is not expected for actual colloid–polymer mixtures, showing that accounting for the influence of polymer physics on (interfacial) properties is essential.

Finally, it should be remarked that several authors [22, 23, 29, 35] have reported on DFT calculations that show the presence of oscillations in the colloid density profiles on the colloidal liquid side, especially near the triple point. The simple square gradient van der Waals approach discussed here cannot reproduce such features. However, the work of Moncho-Jordá et al. [29] suggests that these oscillations are only present in case polymers are described as PHSs, and disappear due to a lowering of the triple point when taking polymer interactions into account. On the other hand, DFT computations of Bryk [23] do predict oscillatory density profiles when the polymers are described as freely jointed tangentially bonded hard-sphere chains. In practice, the presence of capillary waves, which will be discussed in the next section, may further hinder the observation of such oscillations. Therefore, whether these oscillations can be observed in an experimental colloid–polymer mixture remains an open question.

5.3 Some Dynamic Properties of the Colloidal Gas–Liquid Interface

5.3.1 Thermal Capillary Waves

In the previous section it was shown that gas–liquid interfaces in general, and those of colloid–polymer mixtures in particular, are not infinitely sharp but have a finite width, even though, macroscopically, they appear to be sharp. Macroscopically, these interfaces also appear to be flat far from any (solid) surfaces. However, the thermal energy unavoidably distorts the local interface position on a more microscopic level, leading to a corrugated or rough interface on the colloidal length scale.

These fluctuations in the position of the interface are known as thermally excited capillary waves, or (thermal) capillary waves in short. They have been predicted for molecular systems by von Smoluchowski in 1908 [36], and were theoretically quantified by Mandelstam a few years later [37]. Since then, their existence has been confirmed for molecular liquids with light, neutrons and X-rays [38, 39].

The mean-squared interfacial roughness due to thermal fluctuations is given approximately by

$$\begin{aligned} {} \langle h^2 \rangle \sim \frac{kT}{\gamma } = L_\textrm{T}^2. {}\end{aligned}$$

(5.21)

The thermal length $L_\textrm{T} \equiv \sqrt{kT/\gamma }$ for a molecular gas–liquid interface with an interfacial tension of 50 mN/m is of the order of 0.3 nm and cannot be visually observed directly. However, in the case of coexisting colloid–polymer mixtures the interfacial tension can easily reach values as low as 50 nN/m. For these values, the thermal length becomes about 0.3 $\upmu$m and can be quantified using optical techniques.

Another important interfacial aspect that dictates whether these fluctuations may be observed visually is their dynamics. In the viscous hydrodynamic regime, this is governed by the balance between interfacial tension and viscous forces. The typical velocity can be estimated from the quasi-static Stokes equation:

$$\begin{aligned} {} \nabla P = \eta \nabla ^2 v, {}\end{aligned}$$

(5.22)

where P is the pressure, $\eta$ is the viscosity and v is the velocity. The capillary pressure P is proportional to $\gamma /L$, with $\gamma$ the interfacial tension and L a typical length scale. Additionally, the gradient terms ($\nabla$) scale as 1/L. This leads to the definition of the so-called capillary velocity,

$$\begin{aligned} {} v_\textrm{c} = \frac{\gamma }{\eta }. {}\end{aligned}$$

(5.23)

A correlation time $\tau$ can be defined as the time it takes for capillary waves travelling at a velocity $v_\textrm{c}$ to travel the capillary length $\ell _\textrm{c}$:

$$\begin{aligned} {} \tau = \frac{\ell _\textrm{c}}{v_\textrm{c}}. {}\end{aligned}$$

(5.24)

Table 5.5 Characteristic magnitudes for molecular and colloidal systems (assuming colloids of about $d = 0.3$ $\upmu$m)

In Table 5.5, we have collected the characteristic magnitudes of various relevant interfacial quantities for a molecular and a colloidal system. From this table, it becomes clear that, for colloidal systems, not only the magnitude of the thermal capillary waves can be brought into reach of optical techniques but also their velocity and correlation time become accessible through relatively standard microscopy approaches.

A study was performed by Aarts et al. [40], who made a direct visual observation of thermal capillary waves. They used fluorescent PMMA colloidal spheres with $d = 142\mathrm {\ nm}$ dispersed in decalin, to which polystyrene chains with $R_\textrm{g} = 44\mathrm {\ nm}$ were added ($q = 0.6$). The measured phase diagram is shown in Fig. 5.7. The system was further studied using confocal laser scanning microscopy on the state points labelled I–V.

Fig. 5.7

Reprinted with permission from Ref. [40]. Copyright 2004 American Association for the Advancement of Science (AAAS)

Phase diagram of the system studied by Aarts et al. [40] composed of PMMA colloidal spheres ($d = 142\mathrm {\ nm}$) dispersed in decalin, mixed with polystyrene polymers with $R_\textrm{g} = 44\mathrm {\ nm}$ ($q = 0.6$). Samples in a single-phase fluid state ($\times$) and in two-phase gas–liquid coexistence ($\circ$) are indicated, together with the state points ($\bullet$) of Figs. 5.8 and Table 5.6.

Confocal micrographs of the interface at various state points are shown in Fig. 5.8. It is clear that the height of the fluctuations increases closer to the critical point, where the interfacial tension is lower. From the confocal data, the static and dynamic height correlation functions can be determined, which in turn can be used to obtain the interfacial tension. In this way, Aarts et al. found that the interfacial tension of these state points decreases from about 100 nN/m (state point I) to about 1 nN/m (state point V) (see Table 5.6). For a more detailed discussion of these correlation functions, the reader is referred to Ref. [40].

Fig. 5.8

Reprinted with permission from Ref. [40]. Copyright 2004 AAAS

Observation of thermal capillary waves at the colloidal gas–liquid interface at state points I, III, IV and V (top to bottom), as defined in Fig. 5.7. The size of the area displayed in each image is 17.5 $\upmu$m $\times$ 85 $\upmu$m. The yellow points indicate the position of the interface.

5.3.2 Droplet Coalescence

The process of droplet coalescence is frequently observed in everyday life. Whenever two liquid drops, or a liquid drop and bulk liquid, come into contact, coalescence may occur. Coalescence is favourable since it reduces the total interfacial area and is driven by interfacial tension. The phenomenon has been studied at least since the nineteenth century [41]. The breakup of free-surface flows under the influence of surface tension (e.g., the breakup of a liquid jet) has witnessed renewed interest [42]. Notably, significant progress has been made in the study of the hydrodynamic singularities that occur in these problems [43]. In the case of droplet coalescence, three stages may be identified, as illustrated in Fig. 5.9: first the liquid film between two interfaces drains. Subsequently, this film spontaneously ruptures in a single spot, and finally, this spot (‘neck’) grows.

Table 5.6 Interfacial tension for the state points denoted in the phase diagram of Fig. 5.7. Data taken from Ref. [40]

As we have seen in Table 5.5, the capillary velocity for colloidal systems is of the order of micrometres per second, whereas in molecular systems it is tens of metres per second. This has implications for the coalescence of droplets and can be seen through the Reynolds number

$$\begin{aligned} {} Re = \rho v L /\eta , {}\end{aligned}$$

(5.25)

where v is the characteristic velocity and L the characteristic length. For $Re$ larger than $\sim 1$ inertial effects start to dominate. We can assume that $v \sim v_\textrm{c} = \gamma /\eta$; hence, $Re \sim \rho \gamma L /\eta ^2$. Thus, due to the ultra-low interfacial tensions of colloidal systems, it is evident that the viscous hydrodynamic regime $Re < 1$ is significantly expanded compared to molecular systems when droplet coalescence is concerned.

Fig. 5.9

Representation of the various stages of droplet coalescence. Reprinted with permission from Ref. [16]. Copyright Springer Nature 2008

Due to the slow-down of interfacial dynamics in phase-separated colloid–polymer mixtures, it is possible to observe the process of droplet coalescence visually in great detail, as is shown in Fig. 5.10 [40]. This allows for a more complete understanding of the hydrodynamics of droplet coalescence [44]. For instance, the Brownian interfacial fluctuations can be analysed microscopically to quantify the stochastic nature of the film rupture [45].

Fig. 5.10

Observation of the various stages of droplet coalescence. $t = -19$ s and $-5$ s: film drainage; 0 s: film rupture; 4 s: growth of the neck. The scale bar denotes 5 $\upmu$m. Reprinted with permission from Ref. [40]. Copyright 2004 AAAS

Exercise 5.4. Inertia becomes important if the Reynolds number (Eq. (5.25)) becomes larger than $\sim 1$. For $v = v_\textrm{c}$, we can estimate this happens for length scales larger than $L_0$ and time scales larger than $t_0$:

$$\begin{aligned} {} L > L_0 = \frac{\eta ^2}{\rho \gamma } \qquad \text {and}\qquad t > t_0 = \frac{\eta ^3}{\rho \gamma ^2}. {}\end{aligned}$$

(5.26)

(A) Calculate $L_0$ and $t_0$ for molecular and colloidal systems using the values in Table 5.5. Is it realistic to observe viscous coalescence in ordinary molecular liquids?

(B) Aarts et al. [44] observed viscous coalescence in silicon oil with $\eta = 1\mathrm {\ Pa\,s}$ and $\gamma = 20$ mN/m. What are $L_0$ and $t_0$ for this system? What aspect makes the viscous regime observable in this system, and how is that different from colloidal systems?

To conclude, the ultra-low tensions of the interfaces in demixed colloid–polymer systems predicted by Eq. (5.1) have indeed been experimentally observed. We have seen in this chapter that this leads to new and important findings, such as the direct visual observation of capillary waves and the low Reynolds regime in droplet coalescence.