science

Landau–de Gennes theory

Landau–de Gennes theory is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Landau–de Gennes theory rather than just read about it. In short: In physics, Landau–de Gennes theory describes the NI transition, i.e., phase transition between nematic liquid crystals and isotropic liquids, which is based on the classical Landau's theory and was developed by Pierre-Gilles de Gennes in 1969. The phenomenological theory uses the Q {\displaystyle \mathbf {Q} } tensor as an order parameter in expanding the free energy density.

Key takeaways

Landau–de Gennes theory belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Landau–de Gennes theory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Landau–de Gennes theory from memory before moving on to harder problems.

Reference excerpt

In physics, Landau–de Gennes theory describes the NI transition, i.e., phase transition between nematic liquid crystals and isotropic liquids, which is based on the classical Landau's theory and was developed by Pierre-Gilles de Gennes in 1969. The phenomenological theory uses the Q {\displaystyle \mathbf {Q} } tensor as an order parameter in expanding the free energy density.

Mathematical description The NI transition is a first-order phase transition, albeit it is very weak. The order parameter is the Q {\displaystyle \mathbf {Q} } tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotropic liquid phase. We shall consider a uniaxial Q {\displaystyle \mathbf {Q} } tensor, which is defined by

Q = S ( n ⊗ n − 1 3 I ) {\displaystyle \mathbf {Q} =S(\mathbf {n} \otimes \mathbf {n} -{\tfrac {1}{3}}\mathbf {I} )}

where S = S ( T ) {\displaystyle S=S(T)} is the scalar order parameter and n {\displaystyle \mathbf {n} } is the director. The Q {\displaystyle \mathbf {Q} } tensor is zero in the isotropic liquid phase since the scalar order parameter S {\displaystyle S} is zero, but becomes non-zero in the nematic phase. Near the NI transition, the (Helmholtz or Gibbs) free energy density F {\displaystyle {\mathcal {F}}} is expanded about as

F = F 0 + A 2 Q i j Q j i − B 3 Q i j Q j k Q k i + C 4 ( Q i j Q i j ) 2 {\displaystyle {\mathcal {F}}={\mathcal {F}}_{0}+{\frac {A}{2}}Q_{ij}Q_{ji}-{\frac {B}{3}}Q_{ij}Q_{jk}Q_{ki}+{\frac {C}{4}}(Q_{ij}Q_{ij})^{2}}

or more compactly

F = F 0 + A 2 t r Q 2 − B 3 t r Q 3 + C 4 ( t r Q 2 ) 2 {\displaystyle {\mathcal {F}}={\mathcal {F}}_{0}+{\frac {A}{2}}\mathrm {tr} \,\mathbf {Q} ^{2}-{\frac {B}{3}}\mathrm {tr} \,\mathbf {Q} ^{3}+{\frac {C}{4}}(\mathrm {tr} \,\mathbf {Q} ^{2})^{2}}

where ( A , B , C ) {\displaystyle (A,B,C)} are functions of temperature. Near the phase transition, we can expand A ( T ) = a ( T − T ∗ ) + ⋯ {\displaystyle A(T)=a(T-T_{*})+\cdots } , B ( T ) = b + ⋯ {\displaystyle B(T)=b+\cdots } and C ( T ) = c + ⋯ {\displaystyle C(T)=c+\cdots } with ( a , b , c ) {\displaystyle (a,b,c)} being three positive constants. Now substituting the Q {\displaystyle \mathbf {Q} } tensor results in

F − F 0 = a 3 ( T − T ∗ ) S 2 − 2 b 27 S 3 + c 9 S 4 . {\displaystyle {\mathcal {F}}-{\mathcal {F}}_{0}={\frac {a}{3}}(T-T_{*})S^{2}-{\frac {2b}{27}}S^{3}+{\frac {c}{9}}S^{4}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Landau–de Gennes theory

Start with the simplest possible case. Write down what Landau–de Gennes theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Landau–de Gennes theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Landau–de Gennes theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Landau–de Gennes theory

In research: Landau–de Gennes theory appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Landau–de Gennes theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Landau–de Gennes theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Liquid crystals, Phase transitions, Soft matter, so understanding it makes those chapters shorter.
In everyday life: Look for Landau–de Gennes theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Landau–de Gennes theory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Landau–de Gennes theory means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Landau–de Gennes theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Landau–de Gennes theory in simple terms?

Why does Landau–de Gennes theory matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Landau–de Gennes theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Landau–de Gennes theory.