Surfactant

A surfactant is an interfacially active substance that changes surface tension. A concentration gradient therefore generates tangential Marangoni stress.

A cationic surfactant with a sixteen-carbon hydrocarbon tail and a positively charged trimethylammonium head group, accompanied by a ${\mathrm{Br}}^{-}$ counterion. It is an amphiphile that can form aqueous micelles. Its head-group charge has the opposite sign to that of sodium dodecyl sulfate.

An anionic surfactant with a twelve-carbon hydrocarbon tail and a negatively charged sulfate head group, accompanied by a sodium counterion. This separation of hydrophobic and hydrophilic regions makes it an amphiphile. It forms aqueous micelles under suitable concentration and temperature conditions and is used in sodium dodecyl sulfate polyacrylamide gel electrophoresis.

The material derivative of surface concentration combines tangential and normal area change, surface diffusion, and relaxation towards a reservoir concentration $C_0$. The rate $k$ represents exchange; setting $k=0$ recovers transport without exchange. The concentration is measured per interfacial area, so stretching dilutes it.

Consider a nearly spherical bubble of radius $a$ translating with velocity $\mathbf{U}$ through a liquid of dynamic viscosity $\mu$. Neglect internal gas viscosity and use exterior Stokes flow. Let the surfactant concentration be $C=C_0+C^{\prime }$ and linearize surface tension as $\gamma ={\gamma }_0-{\gamma }_1{C}^{\prime }$ with ${\gamma }_1>0$. Surface transport has surface diffusion coefficient $D_s$ and exchange-relaxation rate $k$.

In the bubble's translating frame, write its tangential surface velocity aswhere $\mathbf{n}$ points out of the bubble. The clean-bubble value is $A=-1/2$; a completely immobile interface has $A=0$. The dipolar surfactant distribution is $C^{\prime }=B\mathbf{U}\cdot \mathbf{n}$, withCombining that transport solution with the tangential Marangoni stress balance givesIncreasing $\lambda$ suppresses interfacial motion, whereas stronger diffusion or exchange reduces the concentration contrast and restores mobility. The linearization separately requires $|C^{\prime }|/C_0\ll 1$, or $2|A||\mathbf{U}|a/(k{a}^2+2D_s)\ll 1$. Small Reynolds number and capillary number are also needed for Stokes flow and a nearly spherical shape; a large immobilization parameter alone does not establish those assumptions.

The dipolar normal traction is balanced by buoyant hydrostatic pressure and the concentration-dependent normal capillary stress, in addition to the constant gas pressure and equilibrium Laplace pressure. A degree-one shape perturbation translates a sphere rather than deforming it; the balance fixes the terminal speed. A uniform gas pressure alone cannot balance this mode.

Linear steady transport on a translating sphere gives $B=2A{C}_{0}a/(k{a}^2+2D_s)$. The components of its unit normal vector are degree-one spherical harmonics. The condition $2|A|Ua/(k{a}^2+2D_s)\ll 1$ ensures the concentration perturbation is small relative to $C_0$.

Outward interfacial flow from a thinning region dilutes surfactant there and raises its surface tension. The resulting Marangoni stress pulls fluid back toward the thinning region, opposes further stretching, and slows rupture.

Consider a long-wave film-rupture model whose strong-Marangoni stress asymptotic growth law isHere $h_0$ is the base film thickness, $\mu$ is dynamic viscosity, $\gamma >0$ is the base surface tension, $V>0$ is the model's destabilizing interaction coefficient with dimensions of energy, and $K=k{h}_0$ is dimensionless wavenumber. The precise relation of $V$ to a microscopic interaction depends on the chosen film model.

Set $y=K^2$. Differentiating $y(1-\Gamma y)$ gives $1-2\Gamma y$, so the unique unstable-band maximum isThe second derivative is negative. Since $\Gamma \gg 1$, the selected $K_m\ll 1$ is consistent with the long-wave assumption. The growth timescale is not itself a rupture time or bubble lifetime: a perturbation starting at amplitude $\delta h_0$ must grow to a specified nonlinear scale, giving an additional logarithmic dependence on the initial disturbance, and the base thickness may evolve before that happens.

An insoluble surfactant remains on an interface. Without surface diffusion, its material concentration changes only through the local surface-area dilation rate.

Surface diffusion transports particles along an interface down a concentration gradient. On an isotropic smooth interface with constant diffusivity $D_s$, its flux is $-D_s{\nabla }_{s}C$ and its concentration source is $D_s{\Delta }_{s}C$, using the surface gradient and surface Laplacian.

For concentration $C$ per unit surface area, conservation on a moving material interface givesThe material derivative follows the surface particles. The two dilution terms are tangential expansion and curvature-induced area change from normal velocity $u_n$; the last term is surface diffusion.

For radius $a$ and tangential velocity ${\mathbf{u}}_s={\mathbf{I}}_s\mathbf{Ax}$ with symmetric traceless $\mathbf{A}$, the surface divergence is $-3\mathbf{n}\cdot \mathbf{An}$. At small surface Péclet number $a^2|\mathbf{A}|/D_s$, neglecting advection of the concentration perturbation givesThe result follows because this quadratic is a degree-two spherical harmonic with surface Laplacian $-6/a^2$ times itself and zero spherical mean.

Soap consists of sodium or potassium salts of long-chain fatty acids. Its hydrophilic carboxylate head and hydrophobic hydrocarbon tail make it an amphiphile; saponification of triglycerides produces these salts.

A detergent is a substance or formulation that assists cleaning, commonly through surfactants with water-compatible and oil-compatible molecular regions. These molecules collect at interfaces and can form micelles that help disperse oily material in water. Agitation, concentration and the composition of both the soil and washing solution affect the process.

A simple comparison is to shake an oil-water mixture with and without a suitable detergent and observe how readily droplets disperse and how long they persist. Lowering surface tension can improve wetting, while adsorption and aggregation change how oily material is carried away. Stable dispersion and removal are different steps: the surrounding fluid must still transport the dispersed material.

Detergents can also disrupt a lipid bilayer. In a protease protection assay, removing that membrane barrier tests whether it caused the apparent protection of an enclosed protein.