Surface tensions

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Surface Tensions

Surface tensions arise from the imbalance of molecular forces at an interface.


A net force at an interface implies that work must be done to expand the surface; that is, the surface tensions can be thought of as forces integrated over distances or the changes in energy between a molecule completely surrounded by molecules and a molecule only partially surrounded by others. In the bulk of the liquid, each molecule is pulled equally in all directions by neighbouring liquid molecules, resulting in a net force of zero. At the surface of the liquid, the molecules are pulled inwards by other molecules deeper inside the liquid and are not attracted as intensely by the molecules in the neighbouring medium (be it vacuum, air or another liquid). Therefore, all of the molecules at the surface are subject to an inward force of molecular attraction which is balanced only by the liquid's resistance to compression, meaning there is no net inward force. However, there is a driving force to diminish the surface area, and in this respect a liquid surface resembles a stretched elastic membrane. Thus the liquid squeezes itself together until it has the locally lowest surface area possible.
An example showing surface tension.
If the cohesion energy per molecule is inside a liquid: <math>U</math>
Then the cohesion energy per molecule at the surface is <math>\frac{U}{2}</math>
If the size of a molecule is a then it occupies an area of <math>a^{2}</math>
Therefore the surface tension is of the order <math>\sigma =\frac{U}{2a^{2}}</math>
If the liquid is near its boiling point then <math>U\approx kT</math>
Or the surface tension is about <math>\sigma =\frac{kT}{2a^{2}}</math>

<math>\text{at }25^{0}\text{ }\sigma =\frac{2\times 10^{-21}J}{a^{2}}\text{ For }a=3Ang\text{ }\sigma =20{}^{mJ}\!\!\diagup\!\!{}_{m^{2}}\;</math>

Witten (p. 155) proposes a surface-tension scaling relation, <math>\sim \frac{\alpha }{{kT}/{\delta A}\;}</math>, where <math>\delta A</math> estimates the area of each flexible unit of the liquid:

Witten, Fig. 16.7
While puzzling over this week's problem set (Witten 6.2) I encountered an interesting discrepancy. According to the above figure, water is the next-to-last data point on the 
plot and has a  surface tension of approximately <math>73mJ/m^2</math> at room temperature. I tried to confirm that by actually calculating the surface tension through the formula : 
<math>\alpha = \frac{k_BT}{a^2}</math>.  
Next, I plug in the Boltzmann constant <math>k_B = 1.38\times 10^{-23}J/K</math>, the room temperature <math>T=296K</math> and <math>a \approx 1\AA</math> as an  
approximate  molecular radius of water (seeing that the oxygen-hydrogen bonds in water are of that length). The surface tension that I get is:
<math>\alpha = \frac{k_BT}{a^2}= \frac{1.38\times 10^{-23}J/K \times 296 K}{10^{-20}m^2}= 408 \times 10^{-3}J/m^2  = 408 mJ/m^2</math> 
What's wrong with this picture?! I went online to double-check the surface tension of water at room temp and I'm still getting Witten's <math>72-75mJ/m^2</math> value. But why can't I  
calculate it accurately? This really puzzled me...
Note: First off, this is meant to be more of an order of magnitude way of calculating surface tension from what I understand.  There are two things wrong with this calculation.  
1) this   calculation is valid at the boiling temperature, which for water is on the order of 400K (around 375 K). 2) the size of a water molecule is closer to 3 Angstroms. 
Plugging these in, I get <math> \alpha \approx 62 mJ/m^2</math> 

Hydrogen bonding in water and metallic bonding in metals raise the energies of interaction considerably above kT and hence all have higher surface tensions.

How can you measure surface forces?

Witten proposes in chapter six how to learn about interfacial energy using various instruments.

  • Contact angle micrometer: In simple cases, a shadow of a droplet on a surface is projected and used to determine the contact angle. We also talked about using the sessile drop method to measure contact angle.
  • Spinning Drop Tensiometer and Wilhemeny Plate "produce controlled increases in the interfacial area and measure the associated work" thereby giving you a measure of the interfacial energy.
  • We already looked at the Langmuir trough: In this there are surfactant molecules trapped on one side of a mobile barrier. This barrier is used to compress the molecules, and as this happens a Wilhemeny plate measures the decrease in interfacial tension associated.
  • Scattering techniques are also used quite a bit, but are not often able to probe heterogeneity in surfaces easily. Scattering techniques can include:
    • Ellipsometry to determine the relative amount of surfactant on a surface. Not a way to look at spatial distribution.
    • Evanescent wave fluorescence for looking at spatial distribution at the 10X nanometer scale.
    • x-ray and neutron scattering to observe structure at the nm scale.

An example: Spore ejection

Fungi present an elegant application of surface tension forces when they eject spores. For instance, mushrooms need to launch spores away from the gills on their underside to be carried away by the wind, to produce new mushrooms. The process has four main steps, which are illustrated below (as shown on the Australian National Botanic Gardens website):


The sequence is essentially a conversion from energy stored as surface tension to the kinetic energy of a moving spore: 1) the spore secretes a small amount of sugar molecules, which lead to the condensation of water near the attachment point (2). The resulting droplet of water increases in size (3), until it comes into contact with a thin coating of water on the spore's surface. At that point, surface tension pulls the water in the droplet around the surface of the spore, shifting the center of mass forward and launching the spore away from the gill's surface (4). This process is so effective that accelerations as high as 20,000g are possible in some fungi, as studied with high speed photography [2].

Water striders

The photographes show water striders standing on the surface of a pond. It is clearly visible that their feet cause indentations in the water's surface. And it is intuitively evident that the surface with indentations has more surface area than a flat surface. If surface tension tends to minimize surface area, how is it that the water striders are increasing the surface area?

Recall that what nature really tries to minimize is potential energy. By increasing the surface area of the water, the water striders have increased the potential energy of that surface. But note also that the water striders' center of mass is lower than it would be if they were standing on a flat surface. So their potential energy is decreased. Indeed when you combine the two effects, the net potential energy is minimized. If the water striders depressed the surface any more, the increased surface energy would more than cancel the decreased energy of lowering the insects' center of mass. If they depressed the surface any less, their higher center of mass would more than cancel the reduction in surface energy.

The photos of the water striders also illustrate the notion of surface tension being like having an elastic film over the surface of the liquid. In the surface depressions at their feet it is easy to see that the reaction of that imagined elastic film is exactly countering the weight of the insects.

Water Runner - The Basilisk Lizard

The basilisk is a lizard of the genus Basiliscus. Species include the common basilisk (Basiliscus basiliscus), the green basilisk (Basiliscus plumifrons) and the brown basilisk lizard (Basiliscus vittatus - also known as the striped basilisk). The creatures are native to South America. It has the nickname the "Jesus Christ Lizard" or "Jesus Lizard" because when fleeing from a predator, it can gather sufficient momentum to run on the surface of the water for a brief distance. Basilisks have large hind feet with flaps of skin between each toe, much like the webbing on a frog. These are rolled up when the lizard walks on land; but if the basilisk senses danger, it can open up this webbing to increase the surface area on the water relative to its weight, thus allowing it to run on water for short distances. Smaller basilisks can run about 10-20 meters on the water surface without sinking, and can usually run farther than older basilisks.

All in all, water provides a unique challenge for legged locomotion because it readily yields to any applied force. Previous studies have shown that static stability during locomotion is possible only when the center of mass remains within a theoretical region of stability. Running across a highly yielding surface could move the center of mass beyond the edges of the region of stability, potentially leading to tripping or falling. Yet basilisk lizards are proficient water runners. Juvenile basilisk lizards produce greatest support and propulsive forces during the first half of the step, when the foot moves primarily vertically downwards into the water; they also produce large transverse reaction forces that change from medial (79% body weight) to lateral (37% body weight) throughout the step. These forces may act to dynamically stabilize the lizards during water running.

Scrutinizing the kinematics of running on water, each stride the basilisk takes was divided into three phases based on foot kinematics: the slap, stroke, and recovery phases. The slap phase begins as the foot contacts the water and moves vertically downwards through the water. During the stroke phase, the foot sweeps primarily backwards and medially, ultimately shedding a vortex ring as it transitions into the recovery phase. The recovery phase completes a stride cycle, returning the foot to the start of slap.

Plant Transpiration

Detailed illustration of plant transpiration.

Transpiration is the process by which moisture is carried through plants from roots to small pores on the underside of leaves, where it changes to vapor and is released to the atmosphere. Transpiration is essentially evaporation of water from plant leaves. Transpiration also includes a process called guttation, which is the loss of water in liquid form from the uninjured leaf or stem of the plant, principally through water stomata.

Transpirational pull results ultimately from the evaporation of water from the surfaces of cells in the interior of the leaves. This evaporation causes the surface of the water to recess into the pores of the cell wall. Inside the pores, the water forms a concave meniscus. The high surface tension of water pulls the concavity outwards, generating enough force to lift water as high as a hundred meters from ground level to a tree's highest branches. Transpirational pull only works because the vessels transporting the water are very small in diameter, otherwise cavitation would break the water column. And as water evaporates from leaves, more is drawn up through the plant to replace it. When the water pressure within the xylem reaches extreme levels due to low water input from the roots (if, for example, the soil is dry), then the gases come out of solution and form a bubble - an embolism forms, which will spread quickly to other adjacent cells, unless bordered pits are present (these have a plug-like structure called a torus, that seals off the opening between adjacent cells and stops the embolism from spreading).

The cohesion-tension theory is a theory of intermolecular attraction commonly observed in the process of water traveling upwards (against the force of gravity) through the xylem of plants, which was put forward by John Joly and Henry Horatio Dixon.

Water is a polar molecule due to the high electronegativity of the oxygen atom, which is an uncommon molecular configuration whereby the oxygen atom has two lone pairs of electrons. When two water molecules approach one another they form a hydrogen bond. The negatively charged oxygen atom of one water molecule forms a hydrogen bond with a positively charged hydrogen atom in another water molecule. This attractive force has several manifestations. Firstly, it causes water to be liquid at room temperature, while other lightweight molecules would be in a gaseous phase. Secondly, it (along with other intermolecular forces) is one of the principal factors responsible for the occurrence of surface tension in liquid water. This attractive force between molecules allows plants to draw water from the root (via osmosis) and then through the xylem to the leaf where photosynthesis converts water and carbon dioxide into glucose.

Water is constantly lost by transpiration in the leaf. When one water molecule is lost another is pulled along. Transpiration pull, utilizing capillary action and the inherent surface tension of water, is the primary mechanism of water movement in plants. However, it is not the only mechanism involved. Any use of water in leaves produces forces that causes water to move into them.

More information can be found at: <>

Cell Sorting

Surface tensions have effects on cellular growth. Recent simulations show that when two different cell types are present with different surface interactions, in 3 dimensions, the cells can sort into distinct domains. When enough minority cells are present, untangling does not require the fluctuations and low tensions shown by previous simulations. Coalescencing still requires these factors though, inagreement with previous models. Here are links to the new article and original paper:, Hutson, SM, G Brodland G, J Yang, J and D Viens. Cell Sorting in Three Dimensions: Topology, Fluctuations, and Fluidlike Instabilities. PRL 101, 148105 (2008). Cell sorting is important in cancer spreading and embryolocical development.