LIQUID SURFACES

Microscopic Picture of Liquid SurfaceA surface is not an infinitesimal sharp boundary in the direction of its normal but it has thicknessIf we consider the density normal to the surface, we can observe that within a few molecules, the density decreases from that of the bulk liquid to that of its vapor

Density of Liquid versus the coordinate normal to its surface

Density is only one criterion to define the thickness of an interfaceAnother parameter is the orientation of the moleculesWater molecules at the surface prefer to be oriented with their negative sides out toward the vapor phaseThe orientation fades with increasing distance from the surfaceAt a distance of 1-2 nm the molecules are again randomly oriented.

The surface of a liquid is a very turbulent placeMolecules evaporate from the liquid into the vapor phase and vice versaIn addition, they diffuse into the bulk phase and molecules from the bulk diffuse to the surfaceApproximately :107 water molecules per second hit a surface area of 10 2 (area covered by one water molecule) thus the average time a water molecule remains on the surface is of the order of 0,1 s.

Surface TensiondW = .dA is called surface tension

The surface tension can also be defined by the force F that is required to hold the slider in place and to balance the surface tensional force|F| = 2b or we can also writeF = -dW/dx = -2bThe force is directed to the left while x increase to the right, therefore it has a negative sign.The unit of surface tension is either J/m2 or N/mThe term surface tension is tied to the concept that the surface stays under a tension

Surface Tension of some Liquids

Interpreting in Molecular LevelFor molecules it is energetically favorable to be surrounded by other moleculesMolecules attract each other by different interactions such as van der Waals forces or by hidrogen bondsWithout this attraction there would not be a condensed phase at all, there would be a vapor phaseAt the surface, molecules are only partially surrounded by other molecules and the number of adjacent moleculs is smaller than in the bulk. In order to bring a molecule from the bulk to the surface, work has to be done

With this view can be interpreted as the energy required to bring molecules from inside the liquid to the surface and to create new surface areaTherefore often the term surface energy is used for With this interpretation of the surface tension in mind we immediately realize that has to be positive, otherwise the Gibbs free energy of interaction would be repulsive and all molecules would immediately evaporate into the gas phase

ExampleEstimate the surface tension of cyclohexane from the energy of vaporization vapU = 30.5 kJ/mol at 25oC. The density of cyclohexane is = 773 kg/m3, its molecular weight is M = 84.16 g/mol.For rough estimate we picture the liquid as being arranged in a cubic structure, each molecule is surrounded by 6 nearest neighbor, thus each bond contibutes roughly vapU/6 = 5.08 kJ/mol. At the surface one neighbor and hence one bond is missing. Per mole we therefore estimate a surface tension of 5.08 kJ/mol

To estimate the surface tension we need to know the surface are occupied by one molecule. If the molecules form a cubic structure, the volume of one unit cell is a3 where a is the distance between nearest neighborsThe distance can be calculated from the density a3 = M/ NA ; a = 0.565 nmThe surface area per molecule is a2, for surface energy we estimate : = (vapU)/6NAa2 = 0.0264 J/m2, its close to the experimental value of 0.0247 J/m2.

Equation of Young and LaplaceIf in equilibrium a liquid surface is curved, there is a pressure difference across itThe surface tension tends to minimize the areaThis result in a planar geometry of the surface, in order to curve the surface, the pressure on one side must be larger than on the other side

Rubber MembraneRubber membrane at the end of cylindrical tube. An inner pressure Pi can be applied, which is different than the outside pressure Pa

The Young-Laplace equation relates the pressure difference between the two phase P and the curvature of the surface

R1 and R2 are the two principal radii of curvature. P is also called Laplace pressure, and equation above also referred to as the Laplace equation

Illustration of the curvature of a cylinder and a sphereFor a cylnder of radius r a convenient choice is R1 = r and R2 = so that the curvature is 1/r + 1/ = 1/rFor a sphere with radius R we have R1 = R2 and the curvature is 1/R + 1/R = 2/R

ExampleHow large is the pressure in a spherical bubble with a diameter of 2 mm and a bubble of 20 nm diameter in pure water, compared with the pressure outside?

Fundamental ImplicationsIf we know the shape of a liquid surface we knows its curvature and we can calculate the pressure differenceIn the absence of external fields (e.g. gravity) the pressure is the same everywhere in the liquid; otherwise there would be a flow of liquid to regions of low pressure, thus P is constant and Young-Laplace equation tells us that in this case the surface of the liquid has the same curvature everywhereWith the help of Young-Laplace equation it is possible to caculate the equilibrium shape of a liquid surface (geometry)

Applying the Young-Laplace EquationApplying Young-Laplace equation to simple geometries is usualy obvious at which side the pressure is higherE.g. both inside a bubble and inside a drop, the pressure is higher than outsideBut in other case, it is not so obvious, because the curvature can have an opposite signE.g. a drop hanging between the planar ends of two cylinders, then the two principal curvatures defined byC1 = 1/R1 and C2 = 1/R2 Can have a different sign. We count it positive if the interface is curved towards the liquidThe pressure difference is defined as P = Pliquid - Pgas

Technique to Measure the Surface TensionThe most common technique is to measure optically the contour of a sessile or pendant dropThe measured contour is then fitted with a contour calculated using the Young-Laplace Eq. from this fit surface tension is obtainedThe same method is applied with a pendant or sessile bubbleIn the maximum-bubble-pressure method the surface tension is determined from the value of the pressure which is necessary to pull a bubble out of a capillary against the Laplace pressure

Maximum-Bubble-Pressure Method

Drop-weight Method Here the liquid is allowed to flow out from the bottom of a capillary tube.Drops are formed which detach when they reach a critical dimension, the weight of a drop falling out of a capillary is measuredAs long as the drop is still hanging at the end of the capillary, its weight is more than balanced by the surface tensionA drop falls off when the gravitational force mg determined by the mass of the drop is no longer balanced by the surface tensionmg = 2rc

Release of a Liquid drop from a capillary

Du-Noy tensiometer and Wilhelmy Plate Method

The Kelvin EquationThe subject of the Kelvin equation is the vapor pressure of a liquidTables of vapor pressures for various liquids can be found in common textbooks, these vapor pressures are reported for vapors in thermodynamic equilibrium with liquids having planar surfacesWhen the liquid surface is curved, the vapor pressure changesThe vapor pressure of a drop is higher than that of a plat, planar surfaceIn a bubble the vapor pressure is reducedThe Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid

The cause for this change in vapor pressure is the Laplace pressureThe raised Laplace pressure in a drop causes molecules to evaporate more easilyIn the liquid, which surrounds a bubble, the pressure with respect to the inner part of the bubble is reduced. This makes it more difficult for molecules to evaporate

Applying Kelvin equationDrop in its vapor. The vapor pressure of a drop is higher than that of a liquid with a planar surface. One consequence is that an aerosol of drops (fog) should be unstableTo see this let us assume that we have a box filled with many drops in a gaseous environment, some drops are larger than othersThe small drops have higher vapor pressure than the large drops, hence more liquid evaporates from their surfaceThis tends to condense into larger dropsWithin a population a drops of different sizes, the bigger drops will grow at the expense of the smaller one, these drops will sink down and at the end bulk liquid fills the bottom of the box

Bubble in a liquidFor a bubble, negative sign has to be used because of the negative curvature of the liquid surface

Here r is the radius of the bubbleThe vapor pressure inside a bubble is therefore reducedWhen liquids heated above the boiling point accasionally tiny bubbles are formedInside the bubble the vapor pressure is reduced, the vapor condenses and the bubble collapsesOnly if a bubble larger than a certain critical size is formed, is it more likely to increase in size rather than to collapse

Capillary CondensationKondensasi uap ke dalam kapiler atau pori-pori kecil walaupun dibawah tekanan uap