Understanding crystallization using density
functional theory and fluctuatinghydrodynamics

James F. Lutsko
Center for Nonlinear Phenomena and Complex Systems
Universite Libre de Bruxelles
Work supported by PRODEX project “Influence of Mass Transport and Surface
Growth Processes on Protein Crystal Perfection” contract ESA AO-2004-070.

Example of direct observation of crystal
nucleation

M. Sleutel, JFL, A. E. S. Van Driessche, M. A. Durán-Olivencia,and D. Maes. Observing classical nucleation theory at work by monitoring phase transitions withmolecular precision. Nature Communications, 5:5598, 2014.

Example of direct observation of nucleation

M. Sleutel, JFL, A. E. S. Van Driessche, M. A. Durán-Olivencia,and D. Maes. Observing classical nucleation theory at work by monitoring phase transitions withmolecular precision. Nature Communications, 5:5598, 2014.

Crystallization is a hot topic ...Why?

Lots of applications
Crystallization… occurs throughout physics: snow flakes, the interior of whitedwarfs, dirty plasmas, colloids, determination of the structure ofmolecules;⚫

... is of importance in human health: e.g. it plays a role in several
pathologies such as malaria and of amyloid aggregation leading toAlzheimer’s;⚫

… is important in chemistry and materials science as a means ofpurification from solution;⚫

… is critical in the production of pharmaceuticals wherepolymorphism is a pressing issue;⚫

Why is this problem hard?
Nucleation begins with the formation of a critical cluster which often
consists of only 10’s or 100’s of molecules ➔ small objects
Nucleation is a rare event: protein nucleation experiments often take day oreven weeks before a single crystallite is formed ➔ mixed time scale
And all of this happens somewhere in a macroscopic, 3D volume ➔ mixed lengthscales

Historically, experiment➔ indirect methods (e.g. light-scattering)
Historically, simulations ➔ small volumes, small time scales… until …
The development of rare event simulation techniques beginning in the 1980’s

Development of new nanoscale observational techniques (AFM, cryo-TEM)
The overwhelming interest in protein nucleation

Nucleation Precursors

Experiments andsimulations have shownthat crystal formation is not
as straightforward as once
imagined but the details areelusive.

Theory has not kept pace so
the process remains poorly
understood.
Credit: J. kemsley, Chem. & Eng. News, 93, 28 (2015).

Why nucleation?
Nucleation occurs when there is an extensive free energy barrier
between the homogeneous states:

Uniform transition of the entire system requires macroscopic
amounts of energy and is effectively impossible. So the onlymeans is a nonuniform, local transition.

Classical Nucleation Theory: 1. Energetics
Local change of phase:Δω<0γFree energy of cluster:𝛥𝛺 = 𝛥𝜔𝑉 𝑅 + 𝛾𝑆 𝑅

Classical Nucleation Theory: 2. Dynamics

𝑐𝑡 𝑛 = number of clusters of n molecules (per unit volume ) at time t𝑑𝑐𝑡 𝑛𝜕𝜕𝛽𝛥𝛺 𝑛𝜕=−𝑓 𝑛+𝑓 𝑛
𝑐 𝑛, 𝑡𝑑𝑡
𝜕𝑛𝜕𝑛
𝜕𝑛 𝑡𝐽 ≈ 𝑐1 𝑓 𝑛𝑐

12𝜋𝛥𝛽𝛺′′ 𝑛𝑐 𝑒𝑥𝑝 −𝛥𝛽𝛺 𝑛𝑐

𝑓 𝑛 = attachment rate

Beyond the classical paradigm

CNT is heuristic and was never intended to describe anything except thesimplest liquid-vapour nucleation:
Real clusters are not homogeneous and do not have sharp interfaces

Without sharp interfaces, how can one speak of attachment and detachment?
What about multiple order parameters? What replaces the rate dynamics?
Even in liquid-vapour nucleation, what justifies treating bulk transport andcluster growth separately?

The goal of Mesoscopic Nucleation Theory (MNT) is to:• Treat structure and energetics realistically• Treat dynamics and fluctuations consistently

classical DFT
Formal structure
Local density

ρ 𝐫(ensemble-averaged but not coarse-grained)

External field

ϕ 𝐫(walls, electric field, gravity, ...)

A functional:Λ ρ, ϕ = 𝐹 ρ + නρ 𝐫 ϕ 𝐫 𝑑𝐫

(from models: exactly known in a few caseClassic theorems (Hohenberg-Kohn, Mermin) assure us that minimizingwrt density gives equilibrium density and (grand-canonical) free energyΛ ρ ≥ Λ ρ𝑒𝑞𝑢𝑖𝑙 = Ω

Examples:Ideal gas (exact):
Van der Waals:

Landau, Cahn-Hillard, ...
JFL, Adv. Chem. Phys. 144, 1 (2010).

𝐹𝑖𝑑𝑒𝑎𝑙 ρ = න ρ 𝐫 lnρ 𝐫 − ρ 𝐫
𝐹𝑣𝑑𝑊 ρ = න 𝑓 ρ 𝐫+ 𝐾 𝛻ρ 𝐫𝑑𝐫2𝑑𝐫

classical DFT
Exact result: hard rods

Percus, J. Stat. Phys. 15, 505–511 (1976).

classical DFT
Fundamental Measure Theory
Helmholtz functional:
𝑒𝑥𝐹𝐻𝑆 ρ; 𝑑 = 𝐹 𝑖𝑑 ρ; 𝑑 + 𝐹𝐻𝑆
ρ; 𝑑FMT excess functional:
𝐹𝑀𝑇𝐹𝐻𝑆
ρ; 𝑑with

111=න
Φ 𝐫 +
Φ 𝐫 +
Φ 𝐫
π𝑑 2 12π𝑑 2
24π 3

Φ1 𝐫 = −𝑠 𝐫 ln 1 − η 𝐫

Φ2 𝐫 =𝑠2 𝐫 − 𝑣2 𝐫
1−η 𝐫𝑑𝐫Φ3 𝐫 =? ? ?

Where the fundamental measures are:η 𝐫1𝑑= නΘ− 𝑟12 ρ 𝐫2 𝑑𝐫2
2“volume measure”

𝑠 𝐫1𝐯 𝐫1
𝐓 𝐫1𝑑= නδ− 𝑟122“surface measures”

1𝐫ො12 ρ 𝐫2 𝑑𝐫2𝐫ො12 𝐫ො12

Mesoscopic Nucleation Theory:
1. Energetics and structure: cDFT

Simulation data: P. ten Wolde and Daan Frenkel, J.
Chem. Phys. 109, 9901 (1998).

Calculations: JFL, J. Chem. Phys. 129, 244501 (2008).

Mesoscopic Nucleation Theory:
1. Energetics and structure: cDFT
Interregnum:
The tools of cDFT are developed in the grand canonical ensemble (opensystem, number of particles can change). Useful for nucleation because of large
density changes (e.g. vapor to solid).
In the open system, the critical cluster is a saddle point ➔ minimizing atconstant chemical potential is hard.
However, it is easy to prove that minimizing at fixed particle number (“canonicalensemble”, but not really) gives exactly the same density field and in this case,the gce saddle point becomes a minimum.
This “Grand canonical/canonical duality” is a useful toolMesoscopic Nucleation Theory:
1. Energetics and structure: cDFT

JFL and Julien Lam, Phys. Rev. E 98, 12604 (2018).

Mesoscopic Nucleation Theory:
1. Energetics and structure: cDFT

Mesoscopic Nucleation Theory:
1. Energetics and structure: cDFT

Amorphous and crystalline clusters

Free energy surfaces are not enough!
Gradient descent (« steepest descent », « minimum energypathway ») is often used to « identify » nucleation pathways(and, in other contexts, chemical reaction pathways)
Gradient descent:• Choose a starting point, x.
• Draw a circle (with some small radius) around x• Calculate energy at each point on the circle• Move to the point with the lowest energy
Problem: to draw a circle, we need a metric and where does thiscome from?
Naive solution: solve... or𝜕𝐹 𝑥 𝑦𝑑𝑦 𝑖
=−𝑑𝑡
𝜕𝑦 𝑖

𝑑𝑥 𝑖
𝜕𝐹(𝑥)
=−𝑑𝑡
𝜕𝑥 𝑖?

... So this is just a way to hide theambuguity

Free energy surfaces are not enough!
Gradient descent (« steepest descent », « minimum energy
pathway ») is often used to « identify » nucleation pathways(and, in other contexts, chemical reaction pathways)
Gradient descent:• Choose a starting point, x.
• Draw a circle (with some small radius) around x• Calculate energy at each point on the circle• Move to the point with the lowest energy
Problem: to draw a circle, we need a metric and where does thiscome from?
Solution: some additional physics is needed ... problem-specificdynamics.

Mesoscopic Nucleation Theory:
2. Dynamics
Specialized to: large molecules or colloidal particles moving in a fluidof small molecules (Brownian dynamics, over-damped limit).

Continuity equation (conservation of mass)𝜕𝜌 𝐫, 𝑡+ 𝛻 ⋅ 𝑗 𝐫, 𝑡 = 0
𝜕𝑡where𝑗 = 𝜌𝑣

Momentum equation
𝜕𝑗𝑗𝑗+ 𝛻 ⋅ + 𝛻𝑝 + 𝛻 ⋅ 𝛱 = −𝛾𝑗 +𝜕𝑡𝜌convective

2𝛾𝑘𝐵 𝑇𝜌 𝜉𝑡pressure Dissipitive friction
stressGibbs-Duhem relation

𝑑𝑝 𝐫, 𝑡 = 𝜌 𝐫, 𝑡 𝑑𝜇 𝐫; 𝑡 = 𝜌 𝐫, 𝑡 𝑑𝛿𝐹 𝜌𝛿𝜌 𝐫, 𝑡

JFL, J. Chem. Phys. 135, 161101 (2011); 136, 34509 (2012)White noiseMesoscopic Nucleation Theory:
2. Dynamics
Over-damped limit:𝜕𝜌 𝐫, 𝑡𝛿𝛽𝐹= 𝛾𝑘𝐵 𝑇𝛻 ⋅ 𝜌 𝐫, 𝑡 𝛻−𝛻⋅𝜕𝑡𝛿𝜌 𝐫, 𝑡deterministic𝐹𝜌“Dean-Kawasaki model”

Microscopic DFT modelLow density (ideal gas) limit:

𝐹𝑖𝑑𝑒𝑎𝑙 𝜌 = න𝜕𝜌 𝐫, 𝑡= 𝛾𝑘𝐵 𝑇𝛻 2 𝜌 𝐫, 𝑡 − 𝛻 ⋅𝜕𝑡
Coeff of (tracer) diffusion

2𝛾𝑘𝐵 𝑇𝜌 𝐫, 𝑡 𝜉 𝐫, 𝑡
fluctuating𝜌 𝐫 ln 𝜌 𝐫 − 𝜌 𝐫

2𝐷𝜌 𝐫, 𝑡 𝜉 𝐫, 𝑡𝛾𝑘𝐵 𝑇 = 𝐷𝑑𝐫

Mesoscopic Nucleation Theory:
2. Dynamics
𝜕𝜌 𝐫, 𝑡𝛿𝛽𝐹= 𝐷𝛻 ⋅ 𝜌 𝐫, 𝑡 𝛻−𝛻⋅𝜕𝑡𝛿𝜌 𝐫, 𝑡deterministic
Discretized equation:

2𝐷𝜌 𝐫, 𝑡 𝜉 𝐫, 𝑡
fluctuating

𝜌 𝐫, 𝑡 → 𝜌 𝐼, 𝑡 → 𝑥𝑡𝐼

𝐷𝛻 ⋅ 𝜌 𝐫 → 𝑔𝐼𝐽 (𝑥)𝛻⋅2𝐷𝜌 𝐫 → 𝑞𝑎𝐼 (𝑥)

𝛿𝛽𝐹
𝜕𝐹(𝑥)
→
𝛿𝜌 𝐫
𝜕𝑥 𝐽𝑑𝑥 𝑖
𝜕𝐹 𝑥
𝑖 𝑥 𝜉𝑎 𝑡 ,= −𝑔𝑖𝑗 𝑥+2𝜖𝑞𝑎𝑑𝑡
𝜕𝑥 𝑗𝑗𝑔𝑖𝑗 𝑥 = 𝑞𝑎𝑖 𝑥 𝑞𝑎 𝑥

Mesoscopic Nucleation Theory:
2. Dynamics
𝑑𝑥 𝑖
𝜕𝐹 𝑥
𝑖 𝑥 𝜉 𝑎 𝑡 , 𝑔𝑖𝑗 𝑥 = 𝑞 𝑖 𝑥 𝑞 𝑗 𝑥= −𝑔𝑖𝑗 𝑥+2𝜖𝑞𝑎𝑎𝑎𝑑𝑡
𝜕𝑥 𝑗
For nucleation, we want to know how a system that begins in one homogeneous state,transforms into a final inhomogeneous state, 𝜌𝐴 𝑟, which has a critical cluster. 𝜌𝐵 𝑟,For diffusive, stochastic processes, the probability that the system follows a particular pathway
x(t) from the initial state x(0)=xA to the final state x(T)=xB is known exactly and can be
expressed in the form
1 −2 𝑇𝑃 𝑥 = 𝑁 × 𝑒𝑥𝑝 − 𝜖 න 𝐿 𝑥𝑡 𝑑𝑡
20
In the weak noise limit, the Lagrangian is simply𝑘𝑑𝑥 𝑖
𝜕𝐹𝑑𝑥
𝜕𝐹𝐿𝑥 =+ 𝑔𝑖𝑗 𝑗 𝑔𝑗𝑘+ 𝑔𝑘𝑙 𝑙𝑑𝑡𝑑𝑡
𝜕𝑥
𝜕𝑥

This is equivalent to the Freidlin-Wentzell-Graham theory of large deviations

Mesoscopic Nucleation Theory:
2. Dynamics
𝑑𝑥 𝑖
𝜕𝐹 𝑥
𝑖 𝑥 𝜉 𝑎 𝑡 , 𝑔𝑖𝑗 𝑥 = 𝑞 𝑖 𝑥 𝑞 𝑗 𝑥= −𝑔𝑖𝑗 𝑥+2𝜖𝑞𝑎𝑎𝑎𝑑𝑡
𝜕𝑥 𝑗
In the weak noise limit, here is a simple fluctuation theorem concerning goingfrom a metastable basin to a saddle point:
𝑃 𝑥𝑡= 𝑒𝑥𝑝 −𝛽 𝐹 𝑥 ∗ − 𝐹 𝑥𝐴
𝑃 𝑥−𝑡
Using this, it is easy to show that the most likely path (MLP) passes through thecritical cluster and is determined by solving𝑑𝑥 𝑖
𝜕𝐹 𝑥= ±𝑔𝑖𝑗 𝑥𝑑𝑡
𝜕𝑥 𝑗i.e. by gradient descent with a metric, g , dictated by the dynamics- Gradient descent is not exact (weak noise limit)- In general, metric is far from Euclidean:

𝑔𝑟1 𝑟2 𝜌 = 𝛻1 𝜌 𝒓𝟏 𝛻1 𝛿 𝒓𝟏 − 𝒓𝟐

−1

Bridging length scales: Order Parameters
Reduced description in terms of one or more order parametersρ 𝑟, 𝑡 = ρ 𝑟; 𝑥 1 𝑡 , 𝑥 2 𝑡 , . . . = ρ 𝑟; 𝐱 𝑡

𝑑𝑥 𝑖= −𝐷 ෍ 𝑔𝑖𝑗 𝐱 𝑡
𝑑𝑡𝑗

𝜕β𝐹 𝐱 𝑡𝜕𝑥 𝑗 𝑡𝜕𝑔𝑖𝑗 𝑑𝑒𝑡𝑔
𝑖𝑎 𝐱 𝑡++2𝐷෍𝑞𝜕𝑥 𝑗𝑎𝑑𝑒𝑡𝑔
1𝐪𝐪=𝐠“Kinetic coefficients”

ξ𝑎 𝑡white noiseState-dependent noise(Stratonovich interpretation)𝑟with𝐹 ρ →𝐹 𝐱𝑚 𝑟; x = න 4 π𝑟′2 ρ 𝑟′; 𝐱 𝑑𝑟′
01𝜕𝑚 𝑟; 𝐱 𝜕𝑚 𝑟; 𝐱
𝑔𝑖𝑗 𝐱 = න𝑑𝑟4π𝑟 2 ρ 𝑟; 𝐱
𝜕𝑥 𝑖
𝜕𝑥 𝑗

𝜌 𝑟; 𝑅 = 𝜌0 + (𝜌1 − 𝜌0 )Θ 𝑅 − 𝑟 ➔ reproduces CNT

𝑔𝑖𝑗 𝑔𝑗𝑘 = δ𝑖𝑘

Spherical Symmetrym 𝑅 = න Θ 𝑅 − 𝑟 𝜌 𝒓 𝑑𝒓If the mass within a ball of radius R is
Then one can show that𝜕𝑚𝑡 (𝑅)𝜕𝑚𝑡 𝑅 𝛿𝛽𝐹 𝜌𝑡𝜕𝑚𝑡 𝑅= −𝐷− 2𝐷
𝜉𝑡 (𝑅)𝜕𝑡
𝜕𝑅 𝛿𝑚𝑡 𝑅
𝜕𝑅The exact expression for distance in density-space can be shown to be𝑑 𝜌1 , 𝜌2 =with𝑚 𝑅; 𝜌න

𝑅 𝑚; 𝜌1𝑑𝑚⇔ 𝑅 𝑚; 𝜌∞and not the Euclidean

2− 𝑅 𝑚; 𝜌2𝑑 𝜌1 , 𝜌2 =

න

𝜌1 𝑟 − 𝜌2 𝑟

24𝜋𝑟 2 𝑑𝑟0

Note: changing variables from m(R) to R(m) gives a SDE with state
independent noise ...

Results

LJ pair potential

Effective HS radiusvDW DFT model𝑣LJ 𝑟 = 𝑣LJ 𝑟 − 𝑣min 𝛩 𝑟min − 𝑟 + 𝑤att 𝑟

𝑑 = න 𝑒 −𝛽𝑣 𝑟 𝑑𝑟𝐹 𝜌 = 𝐹𝐹𝑀𝑇 𝜌 + න 𝜌 𝒓𝟏 𝜌 𝒓𝟐 𝑤att 𝑟12 𝑑𝒓𝟏 𝑑𝒓𝟐

Liquid-vapor nucleationMNT Application:Long-wavelength nucleation precursors

Support from simulations: J. F. Lutsko and J. Lam, "Long-wavelengthdensity fluctuations as nucleation precursors", Phys. Rev. E, 101,052122 (2020).
Support from experiment: S. Patrap et al, “Out-of-equilibrium
processes in crystallization of organic-inorganic perovskites during
spin coating”, Nature Comm. 12, 5624 (2021).

Liquid-vapor nucleation

Liquid-vapor nucleation

Liquid-vapor nucleationA Microscopic Theory of Crystallization

JFL, Sci. Adv. 5, eaav7399 (2019)

A Microscopic Theory of Crystallization

A Microscopic Theory of Crystallization

Remarks- Classical Nucleation Theory can be derived from this framework.
-The dynamics could be extended to inertial systems (e.g. hydrodynamics rather thandiffusive dynamics.

- The metric is not a meaningless technicality: in spherically symmetric systems,
the Euclidean distance between two density fields:∞𝑑 𝜌1 , 𝜌2 =

න

𝜌1 𝑟 − 𝜌2 𝑟24𝜋𝑟 2 𝑑𝑟0is incorrect. The one dictated by the dynamics can be worked outexplicitly:𝑚 𝑅; 𝜌𝑚 𝑅; 𝜌 = න 𝛩 𝑅 − 𝑟 𝜌 𝑟 4𝜋𝑟 2 𝑑𝑟𝑑 𝜌1 , 𝜌2 =න𝑅 𝑚; 𝜌1− 𝑅 𝑚; 𝜌22𝑑𝑚

so in terms of this variable, the sde has state independent noise …

⇔ 𝑅 𝑚; 𝜌References
JFL, "How crystals form: A theory of nucleation pathways", Science Advances, 5,eaav7399 (2019)
JFL and Julien Lam, "Classical density functional theory, unconstrained crystallization, andpolymorphic behavior", Phys. Rev. E, 98, 12604 (2018)
JFL, "Systematically extending classical nucleation theory", New Journal of Physics,
20, 103015 (2018)JFL and Miguel A. Durán-Olivencia, "A Two-parameter Extension of Classical
Nucleation Theory", J. Phys. Cond. Matt., 27, 235101 (2015)
JFL, "Nucleation of colloids and macromolecules in a finite volume", J. of ChemicalPhysics, 137, 154903 (2012)
JFL, "A dynamical theory of nucleation for colloids and macromolecules", J. ofChemical Physics, 136, 34509 (2012)