Liquid helium is a colourless, transparent, and highly volatile cryogenic liquid. Its density at the normal boiling point is approximately \(0.125\,\text{g cm}^{-3}\).
One of the most remarkable properties of helium is that it does not solidify at atmospheric pressure, even when its temperature approaches absolute zero (0 K). This behaviour arises from the exceptionally weak interatomic forces and the large quantum-mechanical zero-point energy of helium atoms.
Unlike most substances, helium must be subjected to sufficiently high pressure to undergo solidification. For helium, the minimum solidification pressure is approximately 25 bar at temperatures approaching absolute zero.
Another important feature of liquid helium is that it exists in two distinct liquid phases:
- Liquid Helium I (He I): The normal liquid phase, which exists above the lambda-transition temperature.
- Liquid Helium II (He II): The superfluid phase, which exists below the lambda-transition temperature.
At saturated vapour pressure, the transition between these two phases occurs at approximately 2.17 K, known as the lambda point.
Phase Diagram of Helium
The phase diagram of helium represents the variation of pressure with temperature and illustrates the regions in which helium exists as vapour, liquid helium I, liquid helium II, and solid helium.
(i) Liquid–Vapour Equilibrium Curve
The liquid–vapour equilibrium curve, also called the saturated vapour pressure curve, represents the temperatures and pressures at which liquid helium and helium vapour coexist in equilibrium.
The curve terminates at the critical point, corresponding approximately to
Below the critical temperature, helium can exist in either the liquid or vapour phase, depending on the pressure. At atmospheric pressure, the normal boiling point of helium is approximately 4.2 K.
(ii) Solid–Liquid Equilibrium Curve
The solid–liquid equilibrium curve, also known as the fusion curve, represents the pressure and temperature conditions under which solid and liquid helium coexist.
Unlike most substances, helium does not solidify simply by lowering its temperature at atmospheric pressure. At temperatures approaching 0 K, a pressure of approximately 25 bar is required for solidification.
This unusual behaviour is attributed to the large quantum-mechanical zero-point motion of helium atoms, which prevents them from forming a stable crystalline structure at low pressures.
(iii) Absence of a Conventional Triple Point
In most substances, the solid–liquid, liquid–vapour, and solid–vapour equilibrium curves meet at a triple point, where all three phases coexist.
However, helium has no conventional solid–liquid–vapour triple point because its liquid phase remains stable down to extremely low temperatures at sufficiently low pressures.
Thus, the solid–liquid and liquid–vapour equilibrium curves do not intersect.
(iv) Lambda Point
At saturated vapour pressure, liquid helium undergoes a phase transition at approximately
This temperature is called the lambda point. At this temperature, the specific heat exhibits a pronounced anomaly resembling the Greek letter lambda (\(\lambda\)).
The transition is a continuous thermodynamic phase transition associated with the onset of superfluidity.
(v) Lambda Line
The lambda line represents the boundary between liquid helium I and liquid helium II in the pressure–temperature phase diagram. The lambda-transition temperature varies with pressure.
- Above the lambda-transition temperature, helium exists as normal liquid helium I.
- Below the lambda-transition temperature, helium exists as superfluid liquid helium II.
The lambda line extends from the saturated vapour pressure curve to the solid–liquid boundary.
Peculiar Properties of Liquid Helium
(i) Density
The density of liquid helium varies unusually with temperature. When liquid helium I is cooled from its normal boiling point of approximately 4.2 K, its density generally increases as the temperature decreases.
Near the lambda-transition temperature, the density exhibits an anomaly in its temperature dependence. In the superfluid phase, the density varies only slightly with temperature and may exhibit a maximum below the lambda point.
The density of liquid helium near the lambda transition is approximately
The unusual temperature dependence of density reflects the quantum nature of liquid helium and its anomalous thermal expansion.
(ii) Specific Heat
One of the most characteristic properties of liquid helium is the variation of its specific heat with temperature.
When liquid helium I is cooled towards the lambda-transition temperature, its specific heat increases sharply.
At the lambda point, the specific heat exhibits a pronounced singularity. On further cooling into the helium II phase, the specific heat decreases.
The variation resembles the Greek letter lambda (\(\lambda\)), which gives the transition its name.
The specific heat at constant pressure is defined as
where \(H\) is the enthalpy and \(T\) is the absolute temperature.
The pronounced anomaly in \(C_P\) indicates a fundamental change in the thermodynamic behaviour of liquid helium at the lambda transition.
(iii) Viscosity and Superfluidity
The most remarkable property of liquid helium II is its ability to exhibit superfluidity.
In ordinary liquids, viscosity produces resistance to fluid motion. However, below the lambda-transition temperature, helium II contains a superfluid component that can flow without viscous dissipation under suitable conditions.
This behaviour is explained by the two-fluid model, proposed by Tisza and developed by Landau.
According to this model, liquid helium II behaves as a mixture of two interpenetrating components:
- A normal component having finite viscosity and carrying entropy.
- A superfluid component having zero viscosity and carrying no entropy.
The total density is expressed as
where
At the lambda-transition temperature,
As the temperature decreases below the lambda point, the superfluid fraction increases.
In the limit \(T\rightarrow0\),
Thus, at sufficiently low temperatures, nearly the entire liquid behaves as a superfluid.
The superfluid component can pass through extremely narrow capillaries and porous materials with negligible viscous resistance, provided the flow remains below the critical velocity.
This phenomenon is known as superfluidity.
(iv) Thermal Conductivity
Liquid helium II exhibits extraordinarily efficient heat transport compared with ordinary liquids.
In helium I, heat is transported primarily by conventional thermal conduction and fluid motion. However, in helium II, heat transport is strongly influenced by the relative motion of the normal and superfluid components.
According to the two-fluid model, the normal component carries entropy, whereas the superfluid component carries no entropy.
When a temperature gradient is established, the normal and superfluid components can flow in opposite directions, producing an effective transfer of heat with little or no net mass transport.
This phenomenon is called thermal counterflow.
The total mass-current density is
For zero net mass flow,
The heat-current density associated with counterflow is
where \(s\) is the entropy per unit mass, and the expression applies when the net mass current is zero.
Because of this mechanism, helium II can transport heat extremely efficiently and may exhibit an exceptionally large effective thermal conductivity.
However, this effective thermal conductivity depends strongly on temperature, heat flux, and experimental geometry; it is not a single constant material property.
(v) Formation of Rollin Film
One of the most unusual properties of liquid helium II is its ability to spread over and move along solid surfaces in the form of an extremely thin film.
This film is called the Rollin film, named after B. V. Rollin.
The film is typically tens of nanometres thick, although its thickness depends on the surface and experimental conditions.
When a clean solid surface comes into contact with helium II, the liquid can wet the surface and form a thin, mobile film.
The superfluid component can flow through this film with negligible viscous resistance.
Experimental Observations
Consider a glass test tube partially immersed in a bath of liquid helium II.
When the tube is initially empty, a thin film of helium II forms on its outer surface and moves along the walls into the tube. Under suitable conditions, liquid accumulates inside the tube until hydrostatic equilibrium is approached.
Similarly, when a tube containing helium II is partially immersed in a helium II bath, the liquid can move along the surface film, allowing redistribution of liquid between the tube and the bath.
If a tube containing helium II is lifted above the bath, the Rollin film can continue to flow over its rim and along its outer surface. Liquid may accumulate at the bottom of the tube and fall as droplets.
The process can continue until the liquid inside the tube is substantially depleted.
The flow rate depends on factors such as temperature, film thickness, wetted perimeter, surface properties, and the critical velocity of superfluid flow.
Thus, the Rollin-film phenomenon provides direct evidence of the unusual flow properties of superfluid helium.
(vi) Fountain Effect
The fountain effect is another remarkable property of liquid helium II. It was experimentally demonstrated by Allen and Jones in 1938.
When one region of helium II is heated while connected to another region through a superleak, a pressure difference develops between the two regions.
This pressure difference can force liquid helium to emerge from a narrow capillary in the form of a fountain.
Experimental Arrangement
The apparatus consists of a vertical tube whose lower end contains a porous plug made of fine powder or another suitable porous material.
The porous plug acts as a superleak, allowing the superfluid component to pass through while strongly restricting the flow of the normal component.
The upper end of the tube is connected to a narrow capillary nozzle.
The apparatus is immersed in a Dewar flask containing liquid helium II.
Working Principle
When heat is supplied to the liquid helium inside the tube, its temperature becomes slightly higher than that of the surrounding helium II bath.
This temperature difference produces a thermomechanical pressure difference across the porous plug.
The superfluid component flows from the colder bath into the warmer region through the porous plug.
As a result, the pressure inside the tube increases and liquid helium is expelled through the narrow capillary nozzle, producing a fountain.
The pressure difference is governed by the thermomechanical relation
where
Here, the subscript \(\mu\) indicates constant chemical potential.
For a small temperature difference,
provided \(\rho s\) is approximately constant over the temperature interval.
More generally,
The fountain height is related to the pressure difference by
Therefore,
where \(h\) is the equivalent hydrostatic height of the liquid column and \(g\) is the acceleration due to gravity.
The fountain effect demonstrates the coupling between temperature, pressure, and superfluid motion in helium II.
Comparison Between Liquid Helium I and II
| Property | Liquid Helium I | Liquid Helium II |
|---|---|---|
| Temperature range at saturated vapour pressure | 2.17–4.2 K | Below 2.17 K |
| Nature | Normal quantum liquid | Superfluid |
| Viscosity | Finite | Zero for superfluid component |
| Heat transport | Conventional mechanisms | Highly efficient thermal counterflow |
| Rollin film | No superfluid film flow | Exhibits superfluid film flow |
| Fountain effect | Absent | Present |
| Two-fluid behaviour | No superfluid component | Normal and superfluid components |
| Entropy | Carried by the liquid | Carried by the normal component |
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