Granulometría

Given the nanoaquarium is a hermetically sealed device, it is possible to exploit the bowing of the observation window as a means to determine the pressure within the device. The silicon nitride membranes are relatively large, having been defined as a 100 µm square windows, and very thin, with a 50 nm thickness. The deposition

Figure 4: Time sequence of device filling. Empty, clean device (a). Initially filled device by placing solution on inlet associated with the top of the window (b) followed by the collapse of the membranes (c). Notice that liquid can be seen at the top of the window and along the silicon oxide on the left and right. The devices begins to fill the window region again and separate the membranes once solution is placed on the inlet associated with the bottom of the window image (d-g). An air bubble is trapped within the device (g), but then dissolves (h) leaving a filled device with no bulging (i). The device is then clamped resulting in a slight bulge that is irregular due to working pillars riding the two membranes (j).

Figure 5: Optical Image of partially filled device showing collapsed membranes, pillars, liquid, and a vapor bubble. (Same image as Figure 4d.)

process results in very low residual stress (800M P a), which, combined with the size of the membrane, allows for large deflection without rupture. This is a feature that grants great control over the initial clamped pressure within the device. When a loaded device is observed under an optical microscope, variations in thickness due to bowing are clearly visible. Adjusting the clamping pressure via set screws will alter the membrane deflection as it is a function of device pressure. The resulting light fringes, Newton’s rings, are the direct result of constructive and destructive interfer- ence caused by the variations in liquid thickness and indirectly give the thickness of the device. Grogan et. al. [13, 14] solved this optics problem for the nanoaquarium’s materials and geometry in the absence of connecting pillars. If monochromatic light is used to measure the intensity of the bowing window in an optical microscope, the explicit gauge pressure of the device can be determined prior to pumping down within the electron microscope.

The pressure of a sealed device on the bench top is an important aspect of experi- mental design. The set screws can be adjusted to increase or decrease the pressure and the resulting number of fringes. Since multiple scattering events increase with liquid layer thickness, resolution is inversely related to liquid layer thickness. This means that keeping the liquid as thin as possible is optimal for the sake of resolution. How- ever, additional clamping can be useful when studying systems where the formation of a gaseous species is expected, such as during electrochemical experiments. Higher pressures will lead to a larger saturation concentration require for the nucleation of a second phase, allowing for longer experiments before a bubble forms.

The pressure inside the device after it has been pumped down in the EM vacuum, however, will not necessarily be the same as the bench top pressure. One can imagine the sealed nanoaquarium system acting as a piston-cylinder with a non-linear spring

on top. The non-linear spring is the membrane and its mechanical ability to bulge. When the 1 atm of pressure is removed, the new state of the system will depend on the initial state. If the entire device is a liquid phase, say entirely liquid water, and we assume the final pressure does not drop bellow the saturation pressure (no phase change), there will be very little change in volume and pressure as the liquid water is nearly incompressible. However, if a second vapor phase, say trapped air, is present, then the pressure can drop significantly. The same is true in the cases where dissolved gases come out of solution and nucleate to form a second, vapor phase. This ultimately results in the pressure dropping within the device after being exposed to the vacuum. It is important to note that this is mainly due to having a sealed liquid cell. In commercial holders where the imaging chamber is often connected to atmosphere via tubing, the pressure will be maintained near the conditions outside of the microscope.

Imaging the shape of the membrane provides a simple means to estimate the pressure. Fortunately, the entire membrane can be imaged in the STEM, allowing for an in situ estimate of device pressure if we assume that the intensity in the image is linearly related to water thickness and that the intensity near the silicon edge is associated with the liquid thickness of 200 nm. To close this problem, we fit the intensity profile along a primary axis of the thin silicon nitride membrane (as illustrated inFigure 6a) to the Maier-Schneideret. al.[15] solution of the membrane deflection problem. The shape of the square membrane is given by

h(x, y = 0) =ho 1 + 0.401x 2_+y2 a2 + 1.1611 x2_y2 2a4 cosπx 2a cos πy 2a (2.1)

where h is the height, h0 is the maximum deflection at (0,0) or the center of the membrane, a is the membrane half width, and (x, y) are coordinates measured from

the center of the membrane. Figure 6b and c show, respectively, the shape of the membrane at 1 atm applied pressure and a to-scale 3D rendering of how two membranes would look with the thin gap of our device. For convenience in estimating the pressure, we consider one axis of the membrane. In the case of its primary axis or the “midside-to-midside”, the membrane will take the shape of [15]

h(x, y = 0) =ho 1 + 0.401x 2 a2 cosπx 2a. (2.2)

Since the membrane half length is known, we simply need to extract h0 from the fitting.

Figure 6: Bowing Geometry as fitted to membrane deflection. Fitted liquid thickness (solid line) overlaid on intensity measures as scaled to thickness (a). Overall liquid thickness (b). 3D rending of bowing windows (c).

Once we have extracted the maximum height of the membrane, we can estimate the pressure within the device. Maier-Schneider et. al. [15] provide an analytical expression to directly relate the maximum deflection to an applied pressure as

p(h0) = C1 tσ a2h0+C2(ν) tE a4h 3 0, (2.3)

stress in the film (800 M P a) [14], E is Young’s modulus (325 GP a) [14], C1 = 3.45, (2.4) and C2(ν) = 1.994 1−0.271ν 1−ν . (2.5)

The pressure given here is a gauge pressure (pressure drop across the membrane), meaning, if the membrane is completely flat, then the pressure internal to the device is equal to the pressure outside of the device (neglecting any additional forces such as surface tension pulling the membranes together in cases where the window is not fully wetted). In the case of a device loaded within the vacuum chamber of the microscope, we assume that that the microscope chamber pressure (< 10−5 _{torr) is} negligible. For the case illustrated in Figure 6, we find the maximum deflection to be 614 nm associated with a pressure of 0.349 atm.