Editing Fick's laws of diffusion (section)

=== Sorption rate and collision frequency of diluted solute ===
Adsorption, absorption, and collision of molecules, particles, and surfaces are important problems in many fields. These fundamental processes regulate chemical, biological, and environmental reactions. Their rate can be calculated using the diffusion constant and Fick's laws of diffusion especially when these interactions happen in diluted solutions.

Typically, the diffusion constant of molecules and particles defined by Fick's equation can be calculated using the [[Stokes–Einstein equation]]. In the ultrashort time limit, in the order of the diffusion time ''a''<sup>2</sup>/''D'', where ''a'' is the particle radius, the diffusion is described by the [[Langevin equation]]. At a longer time, the [[Langevin equation]] merges into the [[Stokes–Einstein equation]]. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the [[fluctuation-dissipation theorem]] based on the [[Langevin equation]] in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is:<ref>{{cite journal | vauthors = Bian X, Kim C, Karniadakis GE | title = 111 years of Brownian motion | journal = Soft Matter | volume = 12 | issue = 30 | pages = 6331–6346 | date = August 2016 | pmid = 27396746 | pmc = 5476231 | doi = 10.1039/c6sm01153e | bibcode = 2016SMat...12.6331B }}</ref>
: <math> D(t) = \mu \, k_{\rm B} T\left(1-e^{-t/(m\mu)}\right) , </math>
where (all in SI units)
* ''k''<sub>B</sub> is the [[Boltzmann constant]],
* ''T'' is the [[absolute temperature]],
* ''μ'' is the mobility of the particle in the fluid or gas, which can be calculated using the [[Einstein relation (kinetic theory)]],
* ''m'' is the mass of the particle,
* ''t'' is time.
For a single molecule such as organic molecules or [[biomolecule]]s (e.g. proteins) in water, the exponential term is negligible due to the small product of ''mμ'' in the ultrafast picosecond region, thus irrelevant to the relatively slower adsorption of diluted solute.

[[File:Diffusive sorption probability.png|300px|thumb|right|Scheme of molecular diffusion in the solution. Orange dots are solute molecules, solvent molecules are not drawn, black arrow is an example random walk trajectory, and the red curve is the diffusive Gaussian broadening probability function from the Fick's law of diffusion.<ref name = "Pyle-BJNano">{{cite journal | vauthors = Pyle JR, Chen J | title = Photobleaching of YOYO-1 in super-resolution single DNA fluorescence imaging | journal = Beilstein Journal of Nanotechnology | volume = 8 | pages = 2296–2306 | date = 2017-11-02 | pmid = 29181286 | pmc = 5687005 | doi = 10.3762/bjnano.8.229 }}</ref><sup>:Fig. 9</sup>]]

The [[adsorption]] or [[Absorption (chemistry)|absorption]] rate of a dilute solute to a surface or interface in a (gas or liquid) solution can be calculated using Fick's laws of diffusion. The accumulated number of molecules adsorbed on the surface is expressed by the Langmuir-Schaefer equation by integrating the diffusion flux equation over time as shown in the simulated molecular diffusion in the first section of this page:<ref name = "LangmuirSchaefer1937JACS">{{Cite journal| vauthors = Langmuir I, Schaefer VJ | date = 1937 | title = The Effect of Dissolved Salts on Insoluble Monolayers| journal = Journal of the American Chemical Society | volume = 29 | issue = 11 | pages = 2400–2414 | doi = 10.1021/ja01290a091| bibcode = 1937JAChS..59.2400L }}</ref>
: <math> \Gamma= 2AC_b\sqrt{\frac{Dt}{\pi}}.</math>
* {{mvar|A}} is the surface area (m<sup>2</sup>).
* <math>C_b</math> is the number concentration of the adsorber molecules (solute) in the bulk solution (#/m<sup>3</sup>).
* {{mvar|D}} is diffusion coefficient of the adsorber (m<sup>2</sup>/s).
* {{mvar|t}} is elapsed time (s).
* <math> \Gamma </math> is the accumulated number of molecules in unit # molecules adsorbed during the time <math>t</math>.
The equation is named after American chemists [[Irving Langmuir]] and [[Vincent Schaefer]].

Briefly as explained in,<ref name = "WardTordai1946">{{Cite journal| vauthors = Ward AF, Tordai L |date=1946| title = Time-dependence of Boundary Tensions of Solutions I. The Role of Diffusion in Time-effects| journal = Journal of Chemical Physics | volume = 14 | issue = 7| pages = 453–461 | doi = 10.1063/1.1724167| bibcode = 1946JChPh..14..453W}}</ref>
the concentration gradient profile near a newly created (from <math>t=0</math>) absorptive surface (placed at <math>x=0</math>) in a once uniform bulk solution is solved in the above sections from Fick's equation,
:<math> \frac{\partial C}{\partial x} = \frac{C_b}{\sqrt{\pi Dt}}\text{exp} \left (-\frac{x^2}{4Dt} \right ) , </math>
where {{mvar|C}} is the number concentration of adsorber molecules at <math> x, t </math> (#/m<sup>3</sup>).

The concentration gradient at the subsurface at <math>x = 0</math> is simplified to the pre-exponential factor of the distribution
:<math> \left (\frac{\partial C}{\partial x} \right ) _{x = 0} = \frac{C_b}{\sqrt{\pi Dt}} . </math>
And the rate of diffusion (flux) across area <math>A . </math> of the plane is
:<math> \left (\frac{\partial \Gamma }{\partial t} \right ) _{x = 0} = -\frac{DAC_b}{\sqrt{\pi Dt}} . </math>
Integrating over time,
: <math> \Gamma = \int_0^t \left( \frac{\partial \Gamma}{\partial t} \right) _{x = 0} = 2AC_b\sqrt{\frac{Dt}{\pi}} . </math>

The Langmuir–Schaefer equation can be extended to the Ward–Tordai Equation to account for the "back-diffusion" of rejected molecules from the surface:<ref name = "WardTordai1946" /> 
: <math> \Gamma= 2A{C_\text{b}}\sqrt{\frac{Dt}{\pi}} - A\sqrt{\frac{D}{\pi}}\int_0^\sqrt{t}\frac{C(\tau)}{\sqrt{t-\tau}} \, d\tau , </math> 
where <math>C_b</math> is the bulk concentration, <math>C</math> is the sub-surface concentration (which is a function of time depending on the reaction model of the adsorption), and <math>\tau</math> is a dummy variable.

Monte Carlo simulations show that these two equations work to predict the adsorption rate of systems that form predictable concentration gradients near the surface but have troubles for systems without or with unpredictable concentration gradients, such as typical biosensing systems or when flow and convection are significant.<ref name=JixinMCSimuAdsorption/>

[[File:DiffusiveAdsorptionHistory.jpg|thumb|A brief history of the theories on diffusive adsorption.<ref name=JixinMCSimuAdsorption/>]]

A brief history of diffusive adsorption is shown in the right figure.<ref name=JixinMCSimuAdsorption/> A noticeable challenge of understanding the diffusive adsorption at the single-molecule level is the [[fractal]] nature of diffusion. Most computer simulations pick a time step for diffusion which ignores the fact that there are self-similar finer diffusion events (fractal) within each step. Simulating the fractal diffusion shows that a factor of two corrections should be introduced for the result of a fixed time-step adsorption simulation, bringing it to be consistent with the above two equations.<ref name=JixinMCSimuAdsorption/>

A more problematic result of the above equations is they predict the lower limit of adsorption under ideal situations but is very difficult to predict the actual adsorption rates. The equations are derived at the long-time-limit condition when a stable concentration gradient has been formed near the surface. But real adsorption is often done much faster than this infinite time limit i.e. the concentration gradient, decay of concentration at the sub-surface, is only partially formed before the surface has been saturated or flow is on to maintain a certain gradient, thus the adsorption rate measured is almost always faster than the equations have predicted for low or none energy barrier adsorption (unless there is a significant adsorption energy barrier that slows down the absorption significantly), for example, thousands to millions time faster in the self-assembly of monolayers at the water-air or water-substrate interfaces.<ref name = LangmuirSchaefer1937JACS/> As such, it is necessary to calculate the evolution of the concentration gradient near the surface and find out a proper time to stop the imagined infinite evolution for practical applications. While it is hard to predict when to stop but it is reasonably easy to calculate the shortest time that matters, the critical time when the first nearest neighbor from the substrate surface feels the building-up of the concentration gradient. This yields the upper limit of the adsorption rate under an ideal situation when there are no other factors than diffusion that affect the absorber dynamics:<ref name=JixinMCSimuAdsorption/>
: <math> \langle r \rangle = \frac{4}{\pi}A C_b^{4/3}D , </math>
where:
* <math> \langle r \rangle </math> is the adsorption rate assuming under adsorption energy barrier-free situation, in unit #/s,
* <math> A </math> is the area of the surface of interest on an "infinite and flat" substrate (m<sup>2</sup>),
* <math> C_b </math> is the concentration of the absorber molecule in the bulk solution (#/m<sup>3</sup>),
* <math> D </math> is the diffusion constant of the absorber (solute) in the solution (m<sup>2</sup>/s) defined with Fick's law.

This equation can be used to predict the initial adsorption rate of any system; It can be used to predict the steady-state adsorption rate of a typical biosensing system when the binding site is just a very small fraction of the substrate surface and a near-surface concentration gradient is never formed; It can also be used to predict the adsorption rate of molecules on the surface when there is a significant flow to push the concentration gradient very shallowly in the sub-surface.

This critical time is significantly different from the first passenger arriving time or the mean free-path time. Using the average first-passenger time and Fick's law of diffusion to estimate the average binding rate will significantly over-estimate the concentration gradient because the first passenger usually comes from many layers of neighbors away from the target, thus its arriving time is significantly longer than the nearest neighbor diffusion time. Using the mean free path time plus the Langmuir equation will cause an artificial concentration gradient between the initial location of the first passenger and the target surface because the other neighbor layers have no change yet, thus significantly lower estimate the actual binding time, i.e., the actual first passenger arriving time itself, the inverse of the above rate, is difficult to calculate. If the system can be simplified to 1D diffusion, then the average first passenger time can be calculated using the same nearest neighbor critical diffusion time for the first neighbor distance to be the MSD,<ref name = "Pandey-JPCB2024">{{cite journal | vauthors = Pandey S, Gautam D, Chen J | title = Measuring the Adsorption Cross Section of YOYO-1 to Immobilized DNA Molecules | journal = Journal of Physical Chemistry B | volume =  128| pages = 7254–7262
 | date = 2024-07-16 | issue = 29 | pmid =  39014882| pmc = 11286311| doi = 10.1021/acs.jpcb.4c03359 | pmc-embargo-date = July 25, 2025 }}</ref> 
:<math>L = \sqrt{2Dt} , </math> 
where:
*<math>L~=C_b^{-1/3} </math> (unit m) is the average nearest neighbor distance approximated as cubic packing, where <math>C_b</math> is the solute concentration in the bulk solution (unit # molecule / m<sup>3</sup>),
*<math>D</math> is the diffusion coefficient defined by Fick's equation (unit m<sup>2</sup>/s),
*<math>t</math> is the critical time (unit s).
In this critical time, it is unlikely the first passenger has arrived and adsorbed. But it sets the speed of the layers of neighbors to arrive. At this speed with a concentration gradient that stops around the first neighbor layer, the gradient does not project virtually in the longer time when the actual first passenger arrives. Thus, the average first passenger coming rate (unit # molecule/s) for this 3D diffusion simplified in 1D problem, 
:<math> <r> = \frac{a}{t} = 2aC_b^{2/3}D , </math>
where <math> a</math> is a factor of converting the 3D diffusive adsorption problem into a 1D diffusion problem whose value depends on the system, e.g., a fraction of adsorption area <math>A</math> over solute nearest neighbor sphere surface area <math>4 \pi L^2 /4</math> assuming cubic packing each unit has 8 neighbors shared with other units. This example fraction converges the result to the 3D diffusive adsorption solution shown above with a slight difference in pre-factor due to different packing assumptions and ignoring other neighbors.

When the area of interest is the size of a molecule (specifically, a ''long cylindrical molecule'' such as DNA), the adsorption rate equation represents the collision frequency of two molecules in a diluted solution, with one molecule a specific side and the other no steric dependence, i.e., a molecule (random orientation) hit one side of the other. The diffusion constant need to be updated to the relative diffusion constant between two diffusing molecules. This estimation is especially useful in studying the interaction between a small molecule and a larger molecule such as a protein. The effective diffusion constant is dominated by the smaller one whose diffusion constant can be used instead.

The above hitting rate equation is also useful to predict the kinetics of molecular [[self-assembly]] on a surface. Molecules are randomly oriented in the bulk solution. Assuming 1/6 of the molecules has the right orientation to the surface binding sites, i.e. 1/2 of the z-direction in x, y, z three dimensions, thus the concentration of interest is just 1/6 of the bulk concentration. Put this value into the equation one should be able to calculate the theoretical adsorption kinetic curve using the [[Langmuir adsorption model]]. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry.

[[File:JChen2022JPCA.png|thumb|Comparing collision theory and diffusive collision theory.<ref name=JChen2022JPCA/>]]

The bimolecular collision frequency related to many reactions including protein coagulation/aggregation is initially described by [[Smoluchowski coagulation equation]] proposed by [[Marian Smoluchowski]] in a seminal 1916 publication,<ref name=Smoluchowski1916>{{cite journal | vauthors = Smoluchowski M | title = Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen | journal =  Zeitschrift für Physik| year = 1916 | volume = 17 | pages = 557–571, 585–599 | language = German | bibcode = 1916ZPhy...17..557S }}</ref> derived from [[Brownian motion]] and Fick's laws of diffusion. Under an idealized reaction condition for A + B → product in a diluted solution, Smoluchovski suggested that the molecular flux at the infinite time limit can be calculated from Fick's laws of diffusion yielding a fixed/stable concentration gradient from the target molecule, e.g. B is the target molecule holding fixed relatively, and A is the moving molecule that creates a concentration gradient near the target molecule B due to the coagulation reaction between A and B. Smoluchowski calculated the collision frequency between A and B in the solution with unit #/s/m<sup>3</sup>:
: <math> Z_{AB} = 4{\pi}RD_rC_AC_B,</math>
where:
* <math>R</math> is the radius of the collision,
* <math>D_r = D_A + D_B</math> is the relative diffusion constant between A and B (m<sup>2</sup>/s),
* <math>C_A</math> and <math>C_B</math> are number concentrations of A and B respectively (#/m<sup>3</sup>).

The reaction order of this bimolecular reaction is 2 which is the analogy to the result from [[collision theory]] by replacing the moving speed of the molecule with diffusive flux. In the collision theory, the traveling time between A and B is proportional to the distance which is a similar relationship for the diffusion case if the flux is fixed.

However, under a practical condition, the concentration gradient near the target molecule is evolving over time with the molecular flux evolving as well,<ref name=JixinMCSimuAdsorption/> and on average the flux is much bigger than the infinite time limit flux Smoluchowski has proposed. Before the first passenger arrival time, Fick's equation predicts a concentration gradient over time which does not build up yet in reality. Thus, this Smoluchowski frequency represents the lower limit of the real collision frequency.

In 2022, Chen calculates the upper limit of the collision frequency between A and B in a solution assuming the bulk concentration of the moving molecule is fixed after the first nearest neighbor of the target molecule.<ref name=JChen2022JPCA>{{cite journal | vauthors = Chen J | title = Why Should the Reaction Order of a Bimolecular Reaction be 2.33 Instead of 2? | journal = The Journal of Physical Chemistry A | volume = 126 | issue = 51 | pages = 9719–9725 | date = December 2022 | pmid = 36520427 | pmc = 9805503 | doi = 10.1021/acs.jpca.2c07500 | bibcode = 2022JPCA..126.9719C }}</ref> Thus the concentration gradient evolution stops at the first nearest neighbor layer given a stop-time to calculate the actual flux. He named this the critical time and derived the diffusive collision frequency in unit #/s/m<sup>3</sup>:<ref name=JChen2022JPCA/>
: <math> Z_{AB} = \frac{8}{\pi}{\sigma} D_rC_AC_B\sqrt[3]{C_A+C_B} , </math>
where:
* <math>{\sigma}</math> is the area of the cross-section of the collision (m<sup>2</sup>),
* <math>D_r = D_A + D_B</math> is the relative diffusion constant between A and B (m<sup>2</sup>/s),
* <math>C_A</math> and <math>C_B</math> are number concentrations of A and B respectively (#/m<sup>3</sup>),
* <math>\sqrt[3]{C_A+C_B} </math> represents 1/<d>, where d is the average distance between two molecules.

This equation assumes the upper limit of a diffusive collision frequency between A and B is when the first neighbor layer starts to feel the evolution of the concentration gradient, whose reaction order is {{sfrac|2|1|3}} instead of 2. Both the Smoluchowski equation and the JChen equation satisfy dimensional checks with SI units. But the former is dependent on the radius and the latter is on the area of the collision sphere. From dimensional analysis, there will be an equation dependent on the volume of the collision sphere but eventually, all equations should converge to the same numerical rate of the collision that can be measured experimentally. The actual reaction order for a bimolecular unit reaction could be between 2 and {{sfrac|2|1|3}}, which makes sense because the diffusive collision time is squarely dependent on the distance between the two molecules.

These new equations also avoid the singularity on the adsorption rate at time zero for the Langmuir-Schaefer equation. The infinity rate is justifiable under ideal conditions because when you introduce target molecules magically in a solution of probe molecule or vice versa, there always be a probability of them overlapping at time zero, thus the rate of that two molecules association is infinity. It does not matter that other millions of molecules have to wait for their first mate to diffuse and arrive. The average rate is thus infinity. But statistically this argument is meaningless. The maximum rate of a molecule in a period of time larger than zero is 1, either meet or not, thus the infinite rate at time zero for that molecule pair really should just be one, making the average rate 1/millions or more and statistically negligible. This does not even count in reality no two molecules can magically meet at time zero.