eSFA-Measurements of Confined Cyclohexane

NOTE: The information presented on this page is intended to provide supplementary information [1] about the measurements of structural forces and density variations of confined cyclohexane using the extended surface forces apparatus (eSFA) technique.

  1. M. Heuberger, M. Zäch, N.D. Spencer, SCIENCE, 292, 905, (2001).

Content

Previous SFA Measurements and Interpretation

Cyclohexane and an array of other fluids have already been studied with the conventional SFA in a number of different laboratories. Typically, a drop of cyclohexane is introduced between two sheets of mica and the gap size between the surfaces is succesively reduced. It was found that for very thin films, the molecular 'granularity' of the fluid becomes apparent. In cyclohexane 6-8 film-thickness transitions could be observed that correspond roughly to the expected diameter of the cyclohexane molecule (0.53 nm). Such thin films are found to carry a certain external load (repulsion) before the next film-thickness transition occurs. In the SFA, external loading is done via a compliant spring. The resolution of manual distance measurement in the conventional SFA is in the order of ±0.1-0.3 nm, which makes it difficult to quantify the transitions or follow their time-evolution. When the direction of the approach actuator is reversed, the system is unloaded.

It is commonly found that the external force, F, can be reduced below the value at which the transition occured during the loading cycle. Nevertheless, at some lower overall load, the surfaces jump apart. This jump is due to the unavoidable instability, when measuring forces via spring deflection. The commonly accepted interpretation of these results is that there is a diffuse layering of cyclohexane molecules and the time-averaged mean field forces induced by this layering have an oscillatory shape (Figure 1, blue graph).

Enlarged view: Figure 1
Figure 1: Schematic representation of the measurement of so-called oscillatory forces in cyclohexane. Instability jumps are indicated with lightly colored arrows. Due to spring instabilities, each minimum must be measured in a separate loading/unloading cycle. Such instabilities can occur with any surface potential, if the gradient of the potential exceeds the spring constant, k=dF/dD.

Measurement in the eSFA

Using the extended Surface Forces Apparatus, we have remeasured the behavior of cyclohexane under molecular-level confinement. A selection of these results is presented here. To simplify the discussion, we will use the following conventions:

  • advancing: the approach actuator, M, is approaching the surfaces or the external load is increased (when surfaces are in contact).
  • receding: the approach actuator, M, motion is reversed, the external load is reduced (when surfaces are in contact) or the surfaces are separated.
  • force-run: an advancing/receding or loading/unloading cycle performed in the surface forces apparatus to determine the surface force F(D) as a function of the surface separation.

Over 50 force-runs were performed with the eSFA under variable, but well-defined conditions (temperature control, environmental control (e.g. humidity), mica thickness, instrumental drift control, vibration control, closed-loop, constant-velocity actuator control, rigurous timing, high-precision interferometric measurement).

Enlarged view: Figure 2
Figure 2: Comparision between four force-runs. Force-runs A1 and A2 were measured on the same mica surface location with a 2 hours pause between them. Force-runs B1 and B2 were both measured on adifferent mica sheet than in experiments A. Data from the receding branch are not shown.

Figure 2 shows a collection of four different force-runs (only advancing shown). Experiments A1 and A2 were both performed on October 11, 2000 at the same mica surface locations with a 2 hours pause between them. Experiment B1 was performed on August 12, 2000 and experiment B2 on August 15th, 2000. Experiments B1 and B2 were performed on the same mica surface locations. The mica sheets in experiments A and B were prepared from the same mica-cleaving batch, but from different pieces (i.e. different mica thickness and different mica gluing). The surface separation between consecutive experiments was held at about 1µm and instrumental drift was measured for at least 1 hour before and after each experiment. Drift rates were always between 5-50pm/min. The error of distance measurement is 25pm, and comparable to the size of the symbols (filled circles). The surface force is normalized with the effective radius of curvature of the cylinders R=(R1*R2)^0.5. The contact geometry in experiment A was almost perfectly symmetric (R1=R2=17mm), whereas in experiment B a certain asymmetry was present, R1=15mm, R2=19mm. The advancing/receding motions were driven in constant velocity mode, v=±0.1nm/sec (A1, A2, B1) and v=±0.5nm/sec (B2). Automated FECO detection was performed at regular time intervals of 3000±3ms without stopping the approach actuator motion for the measurement. The actuator introduces no additional vibrations and is slow enough not to disturb the spectral evaluation. Advancing/receding experiments under constant velocity conditions (actuator position, M, constantly monitored) are beneficial, in that they suppress most drift-induced systematic errors of force measurement. The errors of the force calculation are smaller than the size of the symbols and the error bars of distance measurement are in the order of 25pm (not shown). All here presented force-runs were recorded at 25.000°C setpoint temperature.

One can readily observe that the mean thickness of the cyclohexane film undergoes discontinous transitions when confined to a few molecular diameters (Ø=0.53nm). Several points could be measured at similar film thicknesses during loading, which means that the confined film can sustain an external load, which was continously increasing at a rate k*v = 1000N/m * 0.1nm/sec = 100nN/sec. This behavior, which was discovered almost two decades ago, is a remarkable deviation from the intuitively accepted bulk-fluid flow behavior. Such forces are commonly called structural forces,referring to the granular structure of the fluid in small volumes. We reproducibly observe that thinner films can carry higher external loads before they undergo the next film thickness transition. Film-thickness transitions do not occur at regular intervals. This detail, which can also be observed in the conventional SFA, was only discussed in early SFA publications about structural forces. Nevertheless, the average film thickness transition is often of the same order as a molecular diameter, which presumeably spurred the concept of confinement-induced diffuse molecular layering. It is however important to bear in mind that a distance measurement is not equivalent to a measurement of the structure in a film. In contrast to the common picture of layers with well-defined, periodic structure (time-averaged meanfield), we have evidence for a remarkeable amount of disorder. The transition observed at D=2nm in figure 2, for example, extends only over 0.35nm. Our high-precision measurements also show that such film thickness transitions occur over a considerable timescale (up to several minutes) and exhibit a roughly exponentional relaxation. We believe that the system is plastically deforming and more evidence for this hypothesis will be given below.

Reproducibility of the Measurements

The difference between force-runs A and B in Figure 2 is too big to be overseen. Neither do the film thickness transitions occur at the same value of D, nor do they occur at the same external load, F/R. On the other hand we commonly observe a remarkable reproduceability of the force-runs when repeated at the same surface location. Such observations, which we commonly made, are an indication that the local surface morphology or local contact mechanics play a major role for the occurrence and shape of structural forces.

Fine-structure and Thermal Fluctuations

In Figure 2 we also observe that the film thickness fluctuates. These fluctuations in D are about 10x bigger than the statistical error of distance measurement in the eSFA. These film-thickness variations form a persistent fine structure in the F/R(D)-representation that has never been observed before. The fine-structure is hidden within the errors of the conventional SFA technique. A magnified view of the finestructure of experiments A1 and A2 is shown in Figure 3 for both advancing and receding.

Enlarged view: Figure 3
Figure 3: Magnified view of the finestructure of experiments A1 and A2 for both advancing and receding

While thermal fluctuations are expected in small volumes (confinement), the exact mechanisms and a theoretical description in the framework of an extended thermodynamics is not yet available. Nevertheless, the observed fluctuations are of remarkable lateral size of several µm, since our optical probe has a diameter of about 1µm. The two loading/unloading cycles that were performed on the same mica location have a striking similarity. Some features of the finestructure seem to be reproduced, for example during the unloading cycle, which is an indication that some parts of the fine-structure are non-thermal in origin. The fluctuations have an amplitude of up to 0.2nm and characteristic time constants between 1-30 seconds. The fine structure is equally visible with D-evaluation and Dn-evaluation. During instances of growing film thickness, the system seemingly does work against the compliance of the confining surfaces. What looks like a negative compliance of cyclohexane may in fact be a local fluctuation, because surface force and film thickness are not measured (averaged) over the same surface areas.

Density Variations and Fine-structure

Using the Dn-evaluation of Fast Spectral Correlation (FSC) Interferometry, we were able to measure the refractive index of confined cyclohexane with unprecedented precision. Cyclohexane belongs to a class of so-called globular molecules, which tend to form a solid crystalline phase in the solid state (SI). Plastic crystals (PC) are build of molecules that are positionally frozen, but free to rotate. In a way, this is the opposite to a liquid crystal (LC), where the molecules are oriented, but free for translational motion. PCs have a low entropy of melting and are optically isotropic. We can use this optical isotropy to directly correlate the measured refractive index (optical density) to a mass density in the film. The following Figure 4 shows the result of such measurements of the average density of confined cyclohexane. Errors of Dn-evaluation are expected to increase with decreasing film thickness, so we must indicate the appropriate error bars.

Our density measurements show that the density of cyclohexane is subject to remarkeable variations. One can readily observe densities that correspond to the densities of gas, gas+liquid, liquid, liquid+solid (SI) in the normal phase diagram of cyclohexane. Since this is an isothermal experiment (T = 25 °C) and the pressure (p = 0-30 atm) is still moderate, it is quite clear that the phase behavior of cyclohexane must be considerably altered under confinement. While certain details of the density measurements under confinement may vary from experiment to experiment, we can find the following four general trends.

  • First, the time-averaged density tends to decrease as the film thickness is reduced to molecular dimensions. This is rather unexpected because it means that more molecules are expelled from the PCA than from simple geometrical confinement (i.e. simple volume reduction).
  • Second, the average density in the PCA is subject to large fluctuations that have characteristic time constants between 1-30 seconds.
  • These density variations are correlated with the above discussed fine-structure of the force-run. The general trend is that a fluctuation of low density is usually coupled with a fluctuation of larger film thickness. This correlation is readily visible in Figure 4 as a small tilt CCW of groups of points (i.e. all points between two film thickness transitions).
  • A statistical analysis (e.g. histogram) of such groups of points between two film thickness transitions reveals that each group consists of preferred modes (i.e. densities and associated film thickness of higher occurence). This can be seen as patterns in the otherwise randomly distributed points in Figure 4.
Enlarged view: Figure 4
Figure 4: A statistical analysis reveals that each group consists of preferred modes (i.e. densities and associated film thickness of higher occurence). This can be seen as patterns in the otherwise randomly distributed points.

While thermal fluctuations are generally expected to be more important in small volumes, the lateral extension of the variations observed here must be on the order of several µm. This is because our optical probe has a diameter of roughly Ø=1µm. Similar characteristic lengths scales are commonly observed near critical points or in supercritical substances.

Non-equilibrium Effects

The following two figures are added here to show a selection of frequently observed non-equilibrium effects. Figure 5 shows the receding branch of experiments A1 and A2, which was not shown in Figure 2 above.

Enlarged view: Figure 5
Figure 5: Selection of frequently observed non-equilibrium effects.

One can readily see that the receding branch exhibits a larger film thickness below an external load, F/R, that corresponds to the film thickness transition during loaing. Also, the slope seems to be different, which is indicative of the presence of non-equilibrium mechanisms. The coincidence of this effect with the point of the film-thickness transition during loading supports the idea that the film-thickness transition itself is mainly a plastic deformation. The density measured in the PCA during the final phase before surface separation is often between 1.0-1.1, which is close to the value expected for vacuum.

A number of different types of non-equilibrium effects is readily revealed by high-precision measurements. Some of these could be observed during experiment B2, which is conducted at a slightly higher velocity than the other experiments. Figure 6 below is an illustration of these additional effects.

Enlarged view: Figure 6
Figure 6: Advancing and receding branches of experiment B2. Receding was performed at two different velocities, i.e. v=5nm/sec from the point of highest external load to zero external load (dotted arrow), and then at v=0.5nm/sec until full separation (i.e. jump-out) of the surfaces occured. Error bars are not shown for clarity, but are similar to those shown in Figure 5.

As we discussed above (Figure 2), film-thickness transitions often exhibit a measurable transition time. In experiment B2, this can readily be seen from the transition taking place at D = 3.6 nm. Sometimes, one can also observe sub-molecular film thickness transitions. The receeding branch of this experiment extends further than the previous film thickness transition during loading. The time scale of this slow film thickening is several minutes. This is a clear sign of a considerable amount of plasticity in this system. The density measured in the PCA during this plastic deformation is near vacuum level, which suggests that the plastically deformed cyclohexane is not in the center of the contact area.

Outlook

The established model to explain structural forces is that cyclohexane molecules form diffuse layers under confinement. Repulsive and attractive forces are thought to arise as time-averaged mean field forces due to n(D) variations. The new information about confined cyclohexane presented here gives new insights. Density variations with amplitudes exceeding simple layering-transitions and an overall density reduction where the external load is highest (repulsive) cannot be explained by the conventional model. Our results rather suggest that the origin of structural forces might be predominantely plastic in nature. If true, then structural forces are not 'oscillatory' and distance-jumps that were commonly attributed to spring instabilities are in fact plastic deformations exhibiting finite time constants.

The mechanisms of the observed density fluctuations are not fully understood yet, but is seems clear that they are not solely an effect of the small volumes probed. As suggested from thermodynamics of small volumes and from other experiments of cyclohexane in porous silica, there is a confinement-induced depression of the phase transitions temperatures and enthalpies. This is thought to be due to the cyclohexane-surface interaction, which becomes predominant in such narrow, single-slit pores. It may be possible to extend the bulk phase diagram of cyclohexane towards small volumes by adding an additional parameter, the reciprocal film thickness 1/D, and thus obtain an extended thermodynamical description down to the film thicknesses used in our experiments.

We believe that additional experimental information is required to answer some of these very interesting questions. An important experimental improvement would be the simultaneous measurement of D and n at many different points throughout the entire contact zone. This is currently being implemented in our laboratory.

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