Surface Tension

Diagram regarding the forces on 3 molecules of liquid. This cardboard clip which was below the h2o level, has risen gently and smoothly. Surface tension prevents the cardboard clip from submerging and from overflowing the blue glass. Surface tension is caused by the attraction between the liquid's molecules by different intermolecular forces. Within the bulk regarding the liquid, each molecule is pulled equally in every direction by neighbouring liquid molecules, resulting in a net force of zero.

At the surface regarding the liquid, the molecules are pulled inwards by other molecules deeper inside the liquid and are not attracted as intensely by the molecules within the neighbouring moderate be it vacuum, space or another liquid. Therefore, all regarding the molecules at the surface are subject to an inward force of molecular attraction that 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. Therefore, the surface region regarding the liquid shrinks until it has the lowest surface region possible. That explains the spherical shapes of h2o droplets.

Another method to view it is that a molecule in contact with a neighbour is in a decreased state of life than if it weren't in contact with a neighbour. The interior molecules all have as many neighbours as they can possibly have. But the boundary molecules have fewer neighbours than interior molecules and are that is why in a higher state of energy. For the liquid to minimize its life state, it should minimize its many boundary molecules and should that is why minimize its surface area. As a result of surface region minimization, a surface shall assume the smoothest shape it can mathematical proof that smooth shapes minimize surface region relies on use regarding the Euleragrange equation.

Since any curvature within the surface shape conclusions in greater area, a higher life shall also result. Consequently the surface shall push return against any curvature in many the similar to method like a ball pushed uphill shall push return to minimize its gravitational potential energy. Effects in everyday life. Water beading on a leaf. Water dripping from a tap.

The effects of surface tension should be seen with ordinary water:. Beading of rain h2o on the surface of a waxed automobile. H2o adheres weakly to wax and strongly to itself, so h2o clusters into drops. Surface tension sends them their near-spherical shape, due to the fact that a sphere has the smallest likely surface region to volume ratio. Formation of drops occurs when a mass of liquid is stretched.

The animation shows h2o adhering to faucet gaining mass until it is stretched to a spot where the surface tension can no detailed bind it to faucet. It then separates and surface tension forms the drop into a sphere. If a stream of h2o were running from the faucet, the stream should break up into drops during its fall. Gravity stretches the stream, then surface tension pinches it into spheres. Flotation of objects denser than h2o occurs when the object is nonwettable and its mass is tiny enough to be borne by the forces arising from surface tension.

Separation of oil and h2o is caused by a tension within the surface between dissimilar liquids. This kind of surface tension is called interface tension, but its physics are the same. Tears of wine is the formation of drops and rivulets on the side of a glass containing an alcoholic beverage. Its cause is a complex interaction between the differing surface tensions of h2o and ethanol. Surface tension is visible in other common phenomena, mostly when sure substances, surfactants, are used to decrease it:.

Soap bubbles have very huge surface parts with very little bulk. Bubbles in pure h2o are unstable. The addition of surfactants, however, can hold a stabilizing effect on the bubbles look Marangoni effect. Notice that surfactants actually reduce the surface tension of h2o by a factor of 3 or more. Emulsions are a kind of solution in which surface tension plays a role.

Tiny fragments of oil suspended in pure h2o shall spontaneously assemble themselves into many larger masses. But the presence of a surfactant sends a decrease in surface tension, which permits stability of minute droplets of oil within the bulk of h2o or vice versa. Diagram shows, in cross-section, a needle floating on the surface of water. Its weight, Fw, depresses the surface, and is balanced by the surface tension forces on neither side, Fs, which are each parallel to water's surface at the points where it contacts the needle. Notice that the horizontal components regarding the 3 Fs arrows spot in opposite directions, so they cancel each other, but the vertical components spot within the similar to direction and that is why sum up to balance Fw.

Surface tension, represented by the symbol is defined as the force along a line of unit length, where the force is parallel to surface but perpendicular to line. One method to picture this is to imagine a flat soap film bounded on one side by a taut thread of length, L. The thread shall be pulled toward the interior regarding the film by a force equal to 2L the factor of 3 is due to the fact that the soap film has 3 sides, hence 3 surfaces. Surface tension is that is why measured in forces per unit length. Its SI unit is newton per metre but the cgs unit of dyne per cm shall also be used.

One dyn or cm corresponds to 0. An equivalent definition, one that is useful in thermodynamics, is work done per unit area. As such, sequential to increase the surface region of a mass of liquid by an amount, A, a quantity of work, A, is needed. This work is stored as potential energy. Consequently surface tension should be also measured in SI system as joules per square metre and within the cgs system as ergs per cm2.

Since mechanical processes try to retrieve a state of minimum potential energy, a free droplet of liquid naturally assumes a spherical shape, which has the minimum surface region for a provided volume. The equivalence of measurement of life per unit region to force per unit length should be proven by dimensional analysis. Water striders, an usual neuston that skims on surface tension. Water striders use surface tension to walk on the surface of a pondydrophobic setae on the tarsi hold the insect afloat while an apical hydrophilic claw penetrates the surface, allowing it to grip the water. The surface regarding the h2o behaves like an elastic film: the insect's feet cause indentations within the water's surface, increasing its surface area.

This represents an increase in potential life through the surface tension regarding the h2o equal to loss of potential life regarding the insect's lowered center of mass. Surface curvature and pressure. Surface tension forces acting on a tiny differential patch of surface. x and y indicate the no. of bend over the dimensions regarding the patch.

Balancing the tension forces with compression leads to Youngaplace equation. If no force acts normal to a tensioned surface, the surface should remain flat. But if the compression on one side regarding the surface differs from compression on the other side, the compression difference times surface region conclusions in a normal force. Sequential for the surface tension forces to cancel the force due to pressure, the surface should be curved. The diagram shows how surface curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to center regarding the patch.

When all the forces are balanced, the resulting equation is known as the Youngaplace equation:. p is the compression difference. Rx and Ry are radii of curvature in each regarding the axes that are parallel to surface. The quantity in parentheses on the right paw side is in fact twice the mean curvature regarding the surface depending on normalisation. Solutions to this equation determine the shape of h2o drops, puddles, menisci, soap bubbles, and all other shapes determined by surface tension for example the shape regarding the impressions that a h2o strider's feet make on the surface of a pond.

The table below shows how the internal compression of a h2o droplet increases with decreasing radius. For not very tiny drops the effect is subtle, but the compression difference becomes enormous when the drop sizes approach the molecular size. Of course, within the limit of a lone molecule the concept becomes meaningless. p for h2o drops of different radii at STP. Liquid surface like a computer.

To locate the shape regarding the minimal surface bounded by some arbitrary shaped frame creating use of strictly mathematical means should be a daunting task. Yet by fashioning the frame out of wire and dipping it in soap-solution, an approximately minimal surface shall appear within the resulting soap-film within seconds. Without a lone calculation, the soap-film arrives at a solution to a complex minimization equation on its own. The reason for this is that the compression difference throughout a fluid interface is proportional to mean curvature, as seen within the Young-Laplace equation. For an reveal soap film, the compression difference is zero, hence the mean curvature is zero, and minimal surfaces have the property of zero mean curvature.

Since no liquid can exist in a thorough vacuum for very long, the surface of any liquid is an interface between that liquid and some other medium. The top surface of a pond, for example, is an interface between the pond h2o and the air. Surface tension, then, is not a property regarding the liquid alone, but a property regarding the liquid's interface with another medium. If a liquid is in a container, then besides the liquid or space interface at its top surface, there shall also be an interface between the liquid and the walls regarding the container. The surface tension between the liquid and space is usually different greater than its surface tension with the walls of a container.

And where the 3 surfaces meet, their geometry should be such that all forces balance. Forces at contact spot shown for contact angle greater than 90 left and fewer than 90 right. Where the 3 surfaces meet, they shape a contact angle,, that is the angle the tangent to surface creates with the solid surface. The diagram to right shows 3 examples. Tension forces are shown for the liquid-air interface, the liquid-solid interface, and the solid-air interface.

The example on the left is where the difference between the liquid-solid and solid-air surface tension,, is fewer than the liquid-air surface tension,, but is nevertheless positive, that is. In the diagram, most the vertical and horizontal forces should cancel exactly at the contact point. The horizontal component of is canceled by the adhesive force,. The more telling balance of forces, though, is within the vertical direction. The vertical component of should exactly cancel the force,.

Some liquid-solid contact angles. Since the forces are in direct proportion to their respective surface tensions, we also have:. is the liquid-solid surface tension,. is the liquid-air surface tension,. is the solid-air surface tension,.

is the contact angle, where a concave meniscus has contact angle fewer than 90 and a convex meniscus has contact angle of greater than 90. This means that consequently the difference between the liquid-solid and solid-air surface tension,, is difficult to measure directly, it should be inferred from the with no problems measured contact angle,, if the liquid-air surface tension,, is known. This similar to relationship exists within the diagram on the right. But in this case we look that due to the fact that the contact angle is fewer than 90, the liquid-solid or solid-air surface tension difference should be negative:. Special contact angles.

Observe that within the special case of a water-silver interface where the contact angle is equal to 90, the liquid-solid or solid-air surface tension difference is exactly zero. Another special case is where the contact angle is exactly 180. H2o with specially prepared Teflon approaches this. Contact angle of 180 occurs when the liquid-solid surface tension is exactly equal to liquid-air surface tension. Methods of measurement.

Surface tension should be measured creating use of the pendant drop method on a goniometer. Because surface tension manifests itself in different effects, it offers a many paths to its measurement. Which method is optimal depends upon the nature regarding the liquid being measured, the conditions below which its tension is to be measured, and the stability of its surface when it is deformed. Du Noy Ring method: The general method used to measure surface or interfacial tension. Wetting properties regarding the surface or interface have little influence on this measuring technique.

Maximum pull exerted on the ring by the surface is measured. A minimized version of Du Noy method uses a tiny diameter metal needle instead of a ring, in combination with an above sensitivity microbalance to record maximum pull. The advantage of this method is that very tiny sample volumes below to little tens of microliters should be measured with very high precision, without the should correct for buoyancy for a needle or rather, rod, with real geometry. Further, the measurement should be performed very quickly, minimally in about 20 seconds. First commercial multichannel tensiometers [CMCeeker] were recently built based on this principle.

Wilhelmy plate method: A universal method mostly suited to confirm surface tension over long time intervals. A vertical plate of known perimeter is attached to a balance, and the force due to wetting is measured. Spinning drop method: This technique is necessary for measuring little interfacial tensions. The diameter of a drop within a heavy phase is measured while most are rotated. Pendant drop method: Surface and interfacial tension should be measured by this technique, even at elevated temperatures and pressures.

Geometry of a drop is analyzed optically. For details, look Drop. Bubble compression method Jaeger's method? A measurement technique for determining surface tension at brief surface ages. Maximum compression of each bubble is measured. Drop volume method: A method for determining interfacial tension like a function of interface age.

Liquid of one density is pumped into a 2nd liquid of an alternate density and time between drops produced is measured. Capillary rise method: The end of a capillary is immersed into the solution. The height at which the solution reaches inside the capillary is related to surface tension by the equation discussed below. Stalagmometric method: A method of weighting and reading a drop of liquid. Sessile drop method: A method for determining surface tension and density by placing a drop on a substrate and measuring the contact angle look Sessile drop technique.

Test ink method: A method for measuring surface tension of substrats creating use of test ink and interpreting the ink reaction. look video VIDEO to display Surface tension measurement. Liquid in a vertical tube. Main article: Capillary action. Diagram of a Mercury Barometer.

An old style mercury barometer consists of a vertical glass tube about 1cm in diameter partially filled with mercury, and with a vacuum called Toricelli's vacuum within the unfilled volume look diagram to right. Notice that the mercury position at the center regarding the tube is higher than at the edges, creating the upper surface regarding the mercury dome-shaped. The center of mass regarding the entire column of mercury should be slightly decreased if the top surface regarding the mercury were flat over the entire crossection regarding the tube. But the dome-shaped top sends slightly fewer surface region to entire mass of mercury. Repeatedly the 3 effects combine to minimize the total potential energy.

Such a surface shape is known like a convex meniscus. The reason we think about the surface region regarding the entire mass of mercury, within the component regarding the surface that is in contact with the glass, is due to the fact that mercury does not adhere at all to glass. So the surface tension regarding the mercury acts over its entire surface area, within where it is in contact with the glass. If instead of glass, the tube were created out of copper, the situation should be very different. Mercury aggressively adheres to copper.

So in a copper tube, the position of mercury at the center regarding the tube shall be decreased rather than higher than at the edges that is, it should be a concave meniscus. In a situation where the liquid adheres to walls of its container, we think about the component regarding the fluid's surface region that is in contact with the container to have negative surface tension. The fluid then works to maximize the contact surface area. So in this case increasing the region in contact with the container decreases rather than increases the potential energy. That decrease is enough to compensate for the increased potential life associated with lifting the fluid near the walls regarding the container.

Illustration of capillary rise and fall. Red=contact angle fewer than 90; blue=contact angle greater than 90. If a tube is sufficiently narrow and the liquid adhesion to its walls is sufficiently strong, surface tension can draw liquid up the tube in a phenomenon known as capillary action. The height the column is lifted to is provided by:. is the height the liquid is lifted,.

is the liquid-air surface tension,. is the density regarding the liquid,. is the radius regarding the capillary,. is the acceleration due to gravity,. is the angle of contact described above.

Note that if is greater than 90, as with mercury in a glass container, the liquid shall be depressed rather than lifted. Profile curve regarding the edge of a puddle where the contact angle is 180. The curve is provided by the formula: where Tiny puddles of h2o on a smooth sleek surface have perceptible thickness. Pouring mercury onto a horizontal flat sheet of glass conclusions in a puddle that has a perceptible thickness. Don't ever try this except below a fume hood.

Mercury vapor is a toxic hazard. The puddle shall spread out only to spot where it is little below 1/2 a centimeter thick, and no thinner. Repeatedly this is due to action of mercury's tough surface tension. The liquid mass flattens out due to the fact that that brings as many regarding the mercury to as little a position as possible. But the surface tension, at the similar to time, is acting to reduce the total surface area.

The result is the compromise of a puddle of a nearly fixed thickness. The similar to surface tension demonstration should be done with water, but only on a surface created of a substance that the h2o does not adhere to. Wax is such a substance. H2o poured onto a smooth, flat, horizontal wax surface, speak a waxed sheet of glass, shall behave similarly to mercury poured onto glass. The thickness of a puddle of liquid on a surface whose contact angle is 180 is provided by:.

is the depth regarding the puddle in centimeters or meters. is the surface tension regarding the liquid in dynes per centimeter or newtons per meter. is the acceleration due to gravity and is equal to 980cm or s2 or 9. is the density regarding the liquid in grams per cubic centimeter or kilograms per cubic meter. Illustration of how decreased contact angle leads to reduction of puddle depth.

In reality, the thicknesses regarding the puddles shall be slightly fewer than what is predicted by the above formula due to the fact that very little surfaces hold a contact angle of 180 with any liquid. When the contact angle is fewer than 180, the thickness is provided by:. For mercury on glass,, , and, which gives. For h2o on paraffin at 25 C,, , and which gives. The formula also predicts that when the contact angle is 0, the liquid shall spread out into a micro-thin layer over the surface.

Such a surface is spoke about to be fully wettable by the liquid. The break up of streams into drops. Intermediate stage of a jet breaking into drops. Radii of curvature within the tensional direction are shown. Equation for the radius regarding the stream is, where is the radius regarding the unperturbed stream, is the amplitude regarding the perturbation, is distance along the axis regarding the stream, and is the wave number.

Main article: Plateauayleigh instability. In day to day life we all observe that a stream of h2o emerging from a faucet shall break up into droplets, no reason how smoothly the stream is emitted from the faucet. This is due to a phenomenon called the Plateauayleigh instability, that is entirely a consequence regarding the effects of surface tension. The explanation of this instability begins with the existence of tiny perturbations within the stream. These are always present, no reason how smooth the stream is.

If the perturbations are resolved into sinusoidal components, we locate that some components grow with time while others decay with time. Between those that grow with time, some grow at faster rates than others. Whether a component decays or grows, and how fast it grows is entirely a function of its wave no. a measure of how many peaks and troughs per centimeter and the radius regarding the original cylindrical stream. As stated above, the mechanical work wanted to increase a surface is.

Hence at constant heat and pressure, surface tension equals Gibbs free life per surface area:. where is Gibbs free life and is the area. Thermodynamics requires that all spontaneous changes of state are accompanied by a decrease in Gibbs free energy. From this it is easy to understand howcome decreasing the surface region of a mass of liquid is always spontaneous , provided it is not coupled to any other life changes. It follows that sequential to increase surface area, a sure no.

of life should be added. Gibbs free life is defined by the equation,, where is enthalpy and is entropy. Based upon this and the fact that surface tension is Gibbs free life per unit area, it is likely to obtain the following expression for entropy per unit area:. Kelvin's Equation for surfaces arises by rearranging the previous equations. It states that surface enthalpy or surface life different from surface free life depends most on surface tension and its derivative with heat at constant compression by the relationship.

Thermodynamics of soap bubble. The compression inside an necessary one surface soap bubble should be derived from thermodynamic free life considerations. At constant heat and particle number, dT = dN = 0, the differential Helmholtz free life is provided by. where P is the difference in compression inside and outside regarding the bubble, and is the surface tension. In equilbrium, dF = 0, and so,.

For a spherical bubble, the volume and surface region are provided basically by. Substituting these relations into the previous expression, we find. which is equivalent to Young-Laplace equation when Rx = Ry. For real soap bubbles, the compression is doubled due to presence of 3 interfaces, one inside and one outside. Influence of temperature.

Temperature dependence regarding the surface tension of pure water. Temperature dependency regarding the surface tension of benzene. Surface tension is dependent on temperature. For that reason, when a price is provided for the surface tension of an interface, heat should be explicitly stated. The general trend is that surface tension decreases with the increase of temperature, reaching a price of 0 at the critical temperature.

For distant details look Etvs rule. There exists only empirical equations to relate surface tension and temperature:. Here V is the molar volume of that substance, TC is the critical heat and k is a constant valid for almost all substances. A typical price is k = 2. 1 x 107 [J K1 mol-2 or 3].

For h2o one can distant use V = 18 ml or mol and TC = 374 C. A variant on Etvs is described by Ramay and Shields:. where the heat offset of seven kelvins sends the formula with an improved fit to reality at decreased temperatures. is a constant for each liquid and n is an empirical factor, whose price is 11 or 9 for organic liquids. This equation was also proposed by van der Waals, who distant proposed that should be provided by the expression,, where is a universal constant for all liquids, and is the critical compression regarding the liquid consequently later experiments located to vary to some degree from one liquid to another.

Both Guggenheim-Katayama and Etvs take into account the fact that surface tension reaches 0 at the critical temperature, whereas Ramay and Shields fails to match reality at this endpoint. Influence of solute concentration. Solutes can have different effects on surface tension depending on their structure:. Little or no effect, for example sugar. Increase surface tension, inorganic salts.

Decrease surface tension progressively, alcohols. Decrease surface tension and, once a minimum is reached, no more effect: surfactants. What complicates the effect is that a solute can exist in an alternate concentration at the surface of a solvent than in its bulk. This difference varies from one solute or solvent combination to another. Gibbs isotherm states that: is known as surface concentration, it represents excess of solute per unit region regarding the surface over what should be present if the bulk concentration prevailed all the method to surface.

It has units of mol or m2. is the concentration regarding the substance within the bulk solution. is the gas constant and the temperature. Certain assumptions are taken in its deduction, that is why Gibbs isotherm can only be applied to necessary very dilute solutions with 3 components. Influence of particle volume on vapour pressure.

See also: Gibbs-Thomson effect. The Clausius-Clapeyron relation leads to another equation also attributed to Kelvin. It explains why, due to the fact that of surface tension, the vapor compression for tiny droplets of liquid in suspension is greater than standard vapor compression of that similar to liquid when the interface is flat. That is to speak that when a liquid is forming tiny droplets, the equilibrium concentration of its vapor in its surroundings is greater. This arises due to the fact that the compression inside the droplet is greater than outside.

Molecules on the surface of a tiny droplet left have, on average, fewer neighbors than those on a flat surface right. Hence they can be bound more weakly to droplet than are flat-surface molecules. is the standard vapor compression for that liquid at that heat and pressure. rk is the Kelvin radius, the radius regarding the droplets. The effect explains supersaturation of vapors.

Within the absence of nucleation sites, tiny droplets should shape prior to they can evolve into larger droplets. This requires a vapor compression many times the vapor compression at the phase transition point. This equation shall also be used in catalyst chemistry to assess mesoporosity for solids. The effect should be viewed in terms regarding the average many molecular neighbors of surface molecules look diagram. The table shows some calculated values of this effect for h2o at different drop sizes:.

P or P0 for h2o drops of different radii at STP. The effect becomes simple for very tiny drop sizes, like a drop of 1nm radius has about 100 molecules inside, that is a quantity tiny enough to need a quantum mechanics analysis. Surface tension of different liquids in dyn or cm against air. Mixture%'s are by weight. dyne or cm shall also be called mN or m milli-Newton per meter in S.

Surface tension, Acetic acid. Capillary wave brief waves on a h2o surface, governed by surface tension and inertia. Cheerio effect the tendency for tiny wettable floating objects to attract one another. Dimensionless numbers. Dortmund Data Bank contains experimental temperature-dependent surface tensions.

Etvs rule a rule for predicting surface tension dependent on temperature. Hydrostatic Equilibrium the effect of gravity pulling reason into a round shape. Meniscus surface curvature formed by a liquid in a container. Mercury beating heart a consequence of inhomogeneous surface tension. Sessile drop technique.

Specific surface life similar to as surface tension in isotropic materials. Stalagmometric method. Surface tension values. Surfactants substances which reduce surface tension. Tears of wine the surface tension induced phenomenon seen on the sides of glasses containing alcoholic beverages.

Tolman length leading term in correcting the surface tension for curved surfaces. Wetting and dewetting. Breakup of a moving sheet of h2o bouncing off of a spoon. Photo of flowing h2o adhering to a hand. Surface tension creates the sheet of h2o between the flow and the hand.

A soap bubble balances surface tension forces against internal pneumatic pressure. Surface tension prevents a coin from sinking: the coin is indisputably denser than water, so it cannot be floating due to buoyancy alone. The entirety regarding the flower lies below the position regarding the undisturbed free surface. The h2o rises smoothly around its edge. Surface tension prevents h2o filling the space between the petals and possibly submerging the flower.

Photo showing the tears of wine phenomenon, that is induced by a combination of surface tension modification of h2o by ethanol together with ethanol evaporating faster than water. A metal cardboard clip floats on water. Multiple can usually be carefully added without overflow of water. An aluminum coin floats on the surface regarding the h2o at 10C. Any extra mass should drop the coin to bottom.

^ a be c d e f g h i j Pierre-Gilles de Gennes; Franoise Brochard-Wyart; David Qur 2002. Capillary and Wetting Phenomena Drops, Bubbles, Pearls, Waves. ^ a be c White, Harvey E. Modern College Physics. MIT Lecture Notes on Surface Tension, lecture six PDF.

Massachusetts Institute of Technology. Retrieved April two 2007. ^ a be c d e f g h i j Sears, Francis Weston; Zemanski, Mark W. University Physics 2nd ed. MIT Lecture Notes on Surface Tension, lecture two PDF.

Massachusetts Institute of Technology. Retrieved April two 2007. MIT Lecture Notes on Surface Tension, lecture 4 PDF. Massachusetts Institute of Technology. Retrieved April two 2007.

NP-Complete Problems and physical reality. a be c d Surface Tension by the Ring Method Du Nouy Method pdf. ^ a be Surface and Interfacial Tension. Langmuir-Blodgett Instruments. ^ Surfacants at interfaces PDF.

Surface Tension physics lecture notes. ^ Sessile Drop Method. ^ a be c d e Moore, Walter J. Physical Chemistry, 3rd ed. ^ a be c d e Adam, Neil Kensington 1941.

The Physics and Chemistry of Surfaces, 3rd ed. Oxford University Press. ^ a be Physical Properties Sources Index: Etvs Constant. Weitkamp; Handbook of heterogeneous catalysis, Vol. 2, page 430; Wiley-VCH; Weinheim; 1997.

^ Lange's Handbook of Chemistry, 10th ed. Wikimedia Commons has press related to: Surface tension. Concise overview of surface tension. On surface tension and interesting real-world cases. MIT Lecture Notes on Surface Tension.

Theory of surface tension measurements. Surface Tensions of Different Liquids. Calculation of temperature-dependent surface tensions for some common components. The Bubble Wall Audio slideshow from the Local High Magnetic Field Lab explaining cohesion, surface tension and hydrogen bonds. General subfields within physics.

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