Young-Laplace equation

The Young-Laplace equation (after Thomas Young and Pierre-Simon Laplace , which they derived independently from each other in 1805) describes the relationship between surface tension , pressure and surface curvature of a liquid . In physiology it is known as Laplace's law and is used there more generally to describe pressures in hollow organs , regardless of whether the force at the interface is due to surface tension.

drops

In a spherical drop of the radius , for example a small drop of water or a gas bubble in a liquid, there is an increased pressure due to the surface tension at the liquid / gas interface : ${\ displaystyle r}$ ${\ displaystyle \ gamma}$ ${\ displaystyle \ Delta p}$

{\ displaystyle \ Delta p = {\ frac {2 \ cdot \ gamma} {r}}}

The smaller the spherical radius, the greater the pressure.

If the radius is reduced to such an extent that it approaches the order of magnitude of the molecular diameter, the surface tension also becomes dependent on the radius, so that this simple equation no longer applies.

Arbitrarily curved surface

If it is not a sphere but an arbitrarily curved surface , the equation is:

{\ displaystyle \ Delta p = \ gamma \ left ({\ frac {1} {r_ {1}}} + {\ frac {1} {r_ {2}}} \ right)}

Here and are the two main radii of curvature at the corresponding point on the surface. ${\ displaystyle r_ {1}}$ ${\ displaystyle r_ {2}}$

Soap bubble

For the pressure inside a soap bubble , the pressure difference is twice as great because the soap skin has two surfaces gas phase / liquid.

Spherical bubbles:

{\ displaystyle \ Delta p = {\ frac {4 \ cdot \ gamma} {r}}}

Non-spherically symmetrical bodies:

{\ displaystyle \ Delta p = 2 \ gamma \ left ({\ frac {1} {r_ {1}}} + {\ frac {1} {r_ {2}}} \ right)}

Multiple bubbles

If there are several soap bubbles nested inside each other, you have to add the sum of the pressure contributions of all soap bubbles that are on the way from the very outside to the cavity under consideration.

This also applies if bubbles stick to one another, such as in bubble chains, layers or a foam package. In the simplest case, 2 identical sized bubbles stick to each other: They are spherical, the partition is flat. If the 2 bubbles involved are of different sizes, they have different internal pressures according to the reciprocal of their radii, the dividing wall bulges under the pressure difference - less, i.e. with a larger radius. If a very small bubble comes into contact with a comparatively much larger one, the pressure in the larger one will be negligibly small and the partition wall will almost exactly complement the spherical shape of the small bubble. Foam packets made up of bubbles of the same size tend to have flat facets as partitions inside, so the chambers will be polyhedra . Since these bubbles snap into place like molecules in a crystal lattice similar to a dense packing of spheres , foam develops a certain rigidity against the deformation that shifts the bubble arrangement.

Derivation

The following applies to the surface of a sphere ${\ displaystyle A}$

{\ displaystyle A = 4 \ pi r ^ {2}}

,

for the volume

{\ displaystyle V = {\ frac {4} {3}} \ pi r ^ {3}}

.

For a small change in the radius to the changes in the surface are ${\ displaystyle \ mathrm {d} r}$

{\ displaystyle \ mathrm {d} A = 8 \ pi r \ mathrm {d} r}

and the volume

{\ displaystyle \ mathrm {d} V = 4 \ pi r ^ {2} \ mathrm {d} r}

.

The work needed to change the surface is with it

{\ displaystyle \ mathrm {d} W = \ gamma \ mathrm {d} A = \ gamma \ cdot 8 \ pi r \ mathrm {d} r}

,

those for changing the volume

{\ displaystyle \ mathrm {d} W = p \ mathrm {d} V = p \ cdot 4 \ pi r ^ {2} \ mathrm {d} r}

.

The formula given above is obtained if the two contributions are equated.

Physiological applications

Alveoli

The air sacs (alveoli) are covered on their inner side with a liquid film, which creates a positive pressure of the alveolar gas mixture with respect to the alveolar wall according to the Laplace law. The alveoli are connected to one another via the bronchial system; If the alveoli lose their radius during exhalation, it would be expected that the pressure difference on the alveolar wall would increase more strongly in the smaller alveoli than in the larger ones: Atelectasis would occur - the small alveoli collapse as they empty into the larger ones. To prevent this from occurring, the liquid film contains surfactant , which keeps the pressure differences constant by reducing the surface tension the less the alveoli are stretched.

The pull that the liquid film exerts on the alveolar walls explains, in addition to the elastic fibers, the tendency of the lungs to contract. In this respect, the surfactant lowers the elastic breathing resistance (= increases the compliance of the lungs) and thus reduces the inspiratory work of breathing.

When premature babies do not produce enough surfactant, the undesirable consequences of Laplace's law lead to respiratory distress syndrome in the newborn .

Aneurysms

The growth of aneurysms can also be explained using Laplace's law: if the blood pressure remains constant , the tension in the vessel wall increases as the radius increases . Since the greater tension leads to further stretching, a vicious circle results , which can lead to the tearing of the vessel wall with life-threatening bleeding .

heart

To understand the relationship between tension in the heart muscle and pressure in the ventricle, it is helpful to think of the heart as a sphere with a radius . If you divide the sphere in half and consider the forces perpendicular to this plane, the equation for the force equilibrium between compressing muscle force and explosive pressure force results ${\ displaystyle r}$

{\ displaystyle {\ begin {aligned} F _ {\ text {muscle}} & = F _ {\ text {pressure}} \\ A _ {\ text {muscle}} \ cdot K & = A _ {\ text {lumen}} \ cdot p \\ 2 \ pi r \ cdot d \ cdot K & = \ pi r ^ {2} \ cdot p \ end {aligned}}}

,

in which stands for the muscle tension (force per cross section) and for the muscle thickness; the thickness is assumed to be small compared to the radius. Adjusting after printing results in a representation of Laplace's law, in which the surface tension is replaced by the product of muscle tension and muscle thickness: ${\ displaystyle K}$ ${\ displaystyle d}$

{\ displaystyle p = {\ frac {2 \ cdot d \ cdot K} {r}}}

This explains why the highest values in the pressure curve of the ventricle are only measured when the myocardium is already quite contracted and the muscle activity has long since exceeded its maximum: With high muscle thickness and low radius, lower muscle tension is sufficient. In the opposite case, if the heart is stretched considerably by filling, it has to respond with a strong contraction in which the excess volume is ejected with increased pressure. However, the stretching reduces the muscle diameter and increases the radius of the heart, so that the necessary pressure cannot be generated with the same muscle tension. The solution to this problem lies in the Frank Starling mechanism : The stretching of the heart muscle makes the contractile apparatus more sensitive to calcium , so that the necessary force is applied.

In cardiac insufficiency , the expansion of the heart muscle using the Frank Starling mechanism represents a compensation mechanism, which, however, damages the heart muscle in the long term and thus leads to progressive dilatation. Even with dilated cardiomyopathy , the unfavorable consequences of Laplace's law can be compensated less and less.

Individual evidence

^ Robert Franz Schmidt , Florian Lang, Manfred Heckmann (eds.): Physiology of humans . 31st edition. Springer Medizin Verlag, Heidelberg 2010, ISBN 978-3-642-01650-9 , chapter 26.2 Frank-Starling mechanism and Laplace's law .

[1] Robert Franz Schmidt , Florian Lang, Manfred Heckmann (eds.): Physiology of humans . 31st edition. Springer Medizin Verlag, Heidelberg 2010, ISBN 978-3-642-01650-9 , chapter 26.2 Frank-Starling mechanism and Laplace's law .