Diffusion

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As a particle travels, the molecule is jostled by collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be the approximation of a random walk. Mathematically, a random walk is a series of steps, one after another, where each step is taken in a completely random direction from the one before. This kind of path was famously analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. In two dimensions this relationship can be written as:

For a list of common mathematical concepts used to model diffusion see: Diffusion Mathematics


Types of Diffusion

SURFACE DIFFUSION -- Model of a single adatom diffusing across a square surface lattice. Note the frequency of vibration of the adatom is greater than the jump rate to nearby sites. Also, the model displays examples of both nearest-neighbor jumps (straight) and next-nearest-neighbor jumps (diagonal). Not to scale on a spatial or temporal basis.
  • Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients
  • Anomalous diffusion in porous medium
  • Atomic diffusion, in solids
  • Eddy diffusion, in coarse-grained description of turbulent flow
  • Effusion of a gas through small holes
  • Electronics|Electronic diffusion, resulting in an current (electricity)|electric current called the diffusion current
  • Facilitated diffusion, present in some organisms
  • Gaseous diffusion, used for isotope separation
  • Heat equation, diffusion of thermal energy
  • Itō diffusion, mathematisation of Brownian motion, continuous stochastic process.
  • Knudsen diffusion of gas in long pores with frequent wall collisions
  • Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
  • Photon diffusion
  • Plasma diffusion
  • Random walk
  • Reverse diffusion, against the concentration gradient, in phase separation
  • Rotational diffusion, random reorientations of molecules
  • Surface diffusion, diffusion of adparticles on a surface
  • Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid


Slot Diffusion

Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.

Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure.

The system includes several reagents (A₁ , A₂ , ... A⩋) on the surface. Their surface concentrations are (c₁ , c₂ , ... c⩋). The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free paces is z=c₀. The sum of all cᵢ (including free places) is constant, the density of adsorption places b.

The jump model gives for the diffusion flux of Aᵢ (i=1,...,n):

Diffusioneq0.png

The corresponding diffusion equation is:

Diffeq1.png

Due to the conservation law, Diffeq2.png and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because Diffeq3.png. For two and more components the equations are nonlinear.

If all particles can exchange their positions with their closest neighbours then a simple generalization gives

Diffeq4.png
Diffeq5.png

where Diffeq6.png is a symmetric matrix of coefficients which characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration c₀.

Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.

Brownian Motion

Brownian Motion describes the stochastic movement of particles as they travel through space. This type of random movement is often referred to as a random walk, which is typical of particles that diffuse about a 1D 2D or 3D space filled with other particles or barriers. To understand Brownian motion, lets start by characterizing this phenomenon in 2D space.
Brownian5.gif
The mathematical description of this process often includes these terms:


D  (Δ): diffusion rate coefficient

  • D = L2 / 2d⋅t
L : step length
d : dimensions
t : time
D (in units: µm²/s) is the mean diffusion rate per unit time, often in µm²/s for biological motion on a molecular level. This refers to how fast, on average, the particle moves along its trajectories. This value is often of ultimate interest, particularly for simulating Brownian motion; however under most circumstances, the diffusion rate cannot be observed directly in empirical experiments - this would require the ability to visualize all the microscopic particles and collisions that dictate the particle's movement, and would have to be done on a nanosecond timescale. In the animation on the right, the diffusion rate can actually be quantified directly; but what is often seen when observing particle diffusion through a microscope would more closely resemble this: FIG:
Brownian5sn.gif{{{2}}} CLICK AWAY FROM IMAGE TO CLOSE
. Instead, D is often calculated from the mean squared diffusion (MSD) path of the particle, defined below.


MSD  (Ω): mean squared displacement

  • MSD = 2d⋅D
MSD (in units: µm²) is the mean squared displacement of a particle over some time period.


K  (κ): standard deviation of the normal distribution for D

  • k = √(d⋅D)
k (in units: µm) is the standard deviation (σ) of the normal distribution of step lengths that, when randomly sampled, will give rise to a diffusion rate D. This value is useful for simulating Brownian motion for a particular diffusion rate.


L  (λ): mean step length

  • L = √(2d⋅D)
L (in units: µm) is the average step length per interval of observation. In diffusion simulations, this is the step size per iteration. This equation sometimes shows up in the equivalent form L = (2d * D).5


Lx  (λˣ): mean 1D step length component of L

  • λˣ = L / √2
Lx or λˣ (in units: µm) is the average 1-dimensional step length for each component (X,Y,Z) dimension of L. For example, simulating 2D particle diffusion will require the generation of individual step lengths for both the X and Y dimension. The total step distance from the origin will be the length of the hypotenuse created by the individual X and Y component step lengths. In fact, the equation: λˣ = L / √2 is derived from the Pythagorean theorem for right triangles, such that 2(λˣ)² = λ² where 2(λˣ)² represents a² + b² and λ² represents c².


Ls  (λˢ): step length scalar coefficient

  • Ls = 1 / √(D/Ds)
Ds : new desired diffusion rate
Ls or λˢ (in units: units) is a coefficient value that, when multiplied by each Lx component step length, will scale those lengths to achieve a new diffusion rate Ds. After scaling, the new diffusion rate = D/(D/Ds).




In statistical mechanics, the mean squared displacement (MSD or average squared displacement) is the most common measure of the spatial extent of random motion; one can think of MSD as the amount of the system "explored" by a random walker.

More Background

The Probability Density Function (PDF) for a particle in one dimension is found by solving the one-dimensional Diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.


As a particle travels, the molecule is jostled by collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be seen to be a good approximation to a random walk. Mathematically, a random walk is a series of steps, one after another, where each step is taken in a completely random direction from the one before. This kind of path was famously analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. In two dimensions this relationship can be written as:

r²=4•D•t

In 1D, since both forward and backward steps are equally probable, we come to the surprising conclusion that the probable distance travelled sums up to zero! This is clearly a useless property to calculate. If however, instead of adding the distance of each step we added the square of the distance, we realise that we will always be adding positive quantities to the total. In this case the sum will be some positive number, which grows larger with every step. This obviously gives a better idea about the distance (squared in this case) that a particle moves. If we assume each step happens at regular time intervals, we can easily see how the square distance grows with time, and Einstein showed that it grows linearly with time.

In a molecular system a molecule moves in three dimensions, but the same principle applies. Also, since we have many molecules to consider we can calculate a square displacement for all of them. The average square distance, taken over all molecules, gives us the mean square displacement. This is what makes the mean square displacement (or MSD for short) significant in science: through its relation to diffusion it is a measurable quantity, one which relates directly to the underlying motion of the molecules.

In molecular dynamics the MSD is easily calculated by adding the squares of the distance. The linear (i.e. straight line) dependence of the MSD plot is apparent. If the slope of this plot is taken, the diffusion coefficient D may be readily obtained.

At very short times however, the plot is not linear. This is because the the path a molecule takes will be an approximate straight line until it collides with its neighbour. Only when it starts the collision process will its path start to resemble a random walk. Until it makes that first collision, we may say it moves with approximately constant velocity, which means the distance it travels is proportional to time, and its MSD is therefore proportional to the time squared. Thus at very short time, the MSD resembles a parabola. This is of course a simplification - the collision between molecules is not like the collision between two pebbles, it is not instantaneous in space or time, but is `spread out' a little in both. This means that the behaviour of the MSD at short time is sometimes more complicated than this MSD plot shows.

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Brownian.gif

Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. This relationship can be written as:

r²=2d•D•t

  • r² MSD
  • d dimensions
  • D diffusion coefficient (diffusion rate)
  • t time step


VIDEO


MEAN SQUARED DISPLACEMENT

Brownian Motion Mean Squared Displacement
  • The goal of this calculation is to relate the simulated particle diffusion to real world values, namely velocity.
  • Particle velocity will be a function of MSD x units ²⁄s which scales on space (units) and time (s) parameters.
  • Space and time in the model are defined arbitrarily as Step_Size and Step where each Step a particle moves a distance randomly chosen from a normal distribution (µ=1,σ=.2)
  • a step size of 1 unit/step will produce a brownian motion MSD of ~0.52 ±0.2 units ²/s
  • empirical observations show that reasonable values for MSD are:
    • PSD 0.01 µm ²/s
    • synaptic 0.05 µm ²/s
    • extrasynaptic 0.1 µm ²/s
  • given an MSD of 0.52 ±0.2 units ²/s at the current parameters: 1 step = 1 unit (at µ=1,σ=.2), the model will need to be scaled such that particles move at an extrasynaptic rate of 0.1 µm ²/s.
  • spines are on average 1 to 10 µm apart, if the model is comparing two spines 1 µm apart, they should be separated by 5 units of model space. This is because the current particle diffusion rate of the model is .5 µm ²/s and the empirical MSD is .1 µm ²/s



to make 0.1 units²step0.1 µm²s. It was found that an XY random step-size of µ=0.4 (σ=.2) units produced an MSE ≈ 0.1 units²step. Then, the arbitrary 0.5 units were given meaning (converted to 0.5 µm) by scaling the model according to real-world values (see below) by making 1 unit = 1 µm; as a convention, a subunit will be 1/10th of a unit, thus 1 subunit = 0.1 µm). The PSD areas were set to 3-subunits (.3 µm) square, 20 subunits (2 µm) apart, within a rectangular field 20 subunits (2 µm) wide and 60 subunits (6 µm) long. Given these scaled dimensions where 10 subunits ≈ 1 µm, a particle with an XY step-size of 0.5 units moving in a straight line, could theoretically go from PSD1 to PSD2 in 4 steps (obviously given the simulated particles are moving with Brownian motion, this lower-bound would be extremely rare).