Salt+solution

=Salt solution=

This page contains the basic physical chemistry of ions solution. Note that in these articles the accent is made on the ion-water solutions, that means that we are talking most of the time about //hydration// (since the solvent is water), but the ideas are also applicable to any solution with arbitrary solvent. In this case //hydration// transforms to //solvation//.

//Requirements from reader:// - basic knowledge of phenomenological thermodynamics (Gibbs free energy, entropy ...) - basic knowledge of statistical mechanics (correlation and distribution functions, ensemble average ...) - basic knowledge of computer simulations and its relation to statistical mechanics (MD).
 * Note**

Table of contents

 * 1) ==Pair correlation function==
 * 2) ==Potential of Mean Force PMF==
 * 3) ==Hydration shell==
 * 4) ==Coordination number==
 * 5) ==Residence time==
 * 6) ==Samoilov energy==
 * 7) ==Positive and negative hydration==
 * 8) ==Hofmeister series==
 * 9) =="Salting in" and "Salting out"==
 * 10) ==Kosmotropes and Chaotropes==
 * 11) ==Collins pyramid==
 * 12) ==Surface tension==
 * 13) ==Solvation Free Energy==
 * 14) ==Debye and Bjerrum length==
 * 15) ==Crystal radii database==
 * 16) ==Water models==

Pair correlation function
Definition, formal derivation of pair correlation function and its connection to [|radial distribution function] are given in formal derivation section in []. For homogeneous system it turns out that the pair correlation function is equal to [|radial distribution function].

Potential of Mean Force (PMF)
Potential of Mean Force can be defined and calculated in different ways. Below you will find three equivalent definitions of PMF:


 * 1).** Let us consider homogeneous system containing N spherically symmetric particles in volume V with temperature T (NVT-ensemble).

The most intuitive way to define the PMF would be as this:


 * PMF is an effective potential acting between two molecules //i// and //j// in liquid phase.**

Let's first consider the interaction between two molecules in gas phase, where we may neglect interactions of these 2 molecules with all others. This interaction results in forces acting on these two particles, since force is minus gradient of the potential. The forces result in accelerations on particles. So, the ith molecule produces force on //j//th molecule, and //j//th molecule - on //i//th molecule.

Now, we would like to use the same scheme for liquid. Consider molecules //i// and //j// at distance //r// out of ensemble of N molecules. Now, the forces acting on //i// and //j// are not only the result of mutual interactions of //i// and //j//, but also are complicated by interaction with all other molecules in ensemble N. Let us formulate this like following: the molecules //i// and //j// interact with each other via modified (compared to gas phase) potential, this potential includes the average effect of all particles onto //i// and //j// molecules. This potential is called - the Potential of Mean Force (PMF). This potential results in the forces acting between the two molecules on distance //r//. And these forces are exactly the average (over ensemble) forces acting on the particles //i// and //j// having occurred at distance //r// between each other in the ensemble of N molecules.

From the definition //1// the PMF can be calculated in the following way. Let us consider the Molecular Dynamics simulation of the system of N spherical particles in regarded ensemble. On each integration step we choose 2 molecules, let say //i// and //j//, calculate the distance //r// between them, and store the force acting on these two molecules (calculated by MD scheme). The we repeat the procedure for sufficient number of MD integration steps. And finally average all the stored forces that we estimated for the same //r// distances in //r//+d//r//. Where d//r// - specified resolution of the PMF calculation. Now, we have the "Mean Force" between two particles in the system as the function of //r//. Integral of the "Mean Force" will give us the "Potential of Mean Force" as the function of //r//.
 * How to calculate:**

An example of PMF calculated with this method see in: Perera and Berkowitz //J.Phys.Chem.// **1993**,//97//,13803

//**Note:**// PMF can be calculated not only as a function of //r//, but actually any spatial variable (f.e. angles) and there combination. It may also be a function of coordinates of several particles.


 * 2).** **PMF can be defined through the n-particle correlation functions.**

Let us define the PMF between 2 non-spherical particles in one-component homogeneous system in T,V,N ensemble :

> math PMF( \mathbf{r}_{ij},\Omega_{ij})= - k_BT \ln g(\mathbf{r}_{ij},\Omega_{ij}) math

Where math g(\textbf{r}_{ij},\Omega_{ij}) math - pair correlation function, math \mathbf{r}_{ij}=\{ x_{ij},y_{ij},z_{ij} \} ~ and ~ \Omega_{ij}=\{ \phi_{ij},\theta_{ij},\psi_{ij} \} math – the radius-vector between ith and jth molecules (f.e. between their center of masses) and the relative orientations of the ith and jth molecules (given by the set of [|Euler angles]).

or equivalently: > math g(\mathbf{r}_{ij},\Omega_{ij}) = e^{ - \beta PMF( \mathbf{r}_{ij},\Omega_{ij})} math

Substituting the expression into definition of pair correlation function (ref?) we get the following: > math -\overrightarrow{grad}_j PMF(\mathbf{r}_{ij},\Omega_{ij}) = \dfrac {\int ... \int exp(-\beta U)(-\overrightarrow{grad}_j U) d\mathbf{r}_3...d\mathbf{r}_Nd\mathbf{\Omega}_3...d\mathbf{\Omega}_N} {\int ... \int exp(-\beta U) d\mathbf{r}_1...d\mathbf{r}_Nd\mathbf{\Omega}_1...d\mathbf{\Omega}_N} math This huge expression is nothing more then just an ensemble average of the - grad_j U - the force acting on particle //j//. Thus PMF is "the potential that gives the mean force acting on particle j, or, i.e., PMF is the //potential of mean force//".

References: D. A. McQuarrie "Statistical Mechanics" 2000, p. 266. A. Ben-Naim, Molecular Theory of Solutions, Oxford University Press, New York, 2006, pp. 56-75.

For homogeneous system containing N spherically symmetric particles in volume V with temperature T (NVT-ensemble) the 2 particle PMF would be:
 * 3).** **“PMF … is the work involved (the Helmholtz energy in the T,V,N ensemble or the Gibbs energy in the T, P, N ensemble) in bringing two (or many) selected particles from infinite separation (in condensed phase) to the final configuration (in condenced phase) …”.**

> math PMF(r) = A(r) - A(\infty) math where A - the Helmholtz free energy.

Reference: A. Ben-Naim, Molecular Theory of Solutions, Oxford University Press, New York, 2006, pp. 56-75.

All the three definitions of PMF are equivalent.

Hydration shell
(See also []). Let us consider an ion dissolved in water. Ion attracts water molecules, ad as the result a number of attracted waters are situated around the ion. The form a "shell" around ion. Rest water molecules are also attracted to ion and to the water molecules in the first "shell". But no more waters can come close to ion because there is no space left. So, they are situated next to the waters in the first shell. They form the second shell molecules around ion. And so on. Until waters at certain distance do not feel the influence of ion and behave as just bulk water molecules. All the water molecules that we decide are distinct from the bulk water form the **hydration shell** around ion, which can be split to **first**, **second** ... **hydration shells** according to its distance from the ion.

Now, the question arises: how to distinguish where the first HS ends and the second starts? The question has no direct answer. One has to introduce some criteria.

The most frequently used criteria is to use ion-water [|radial distribution function] that indicates the relative density of water at a certain distance from ion comparing to bulk density of water. Usually for condensed systems it has an oscillatory view (Figure 1).

Usually the boundary for the first HS is chosen to be the position of first minimum on g(r), for the second - from the first minimum to second minimum on g(r) and so on. But one can implement more sophisticated criteria.

One thing should be noted: all the criteria are artificial. And in MD simulations there is a often question: how to treat molecules that are close to a boundary? Hydration shell is an averaged (thermodynamic) thing. That is why if you have an instant coordinates from your MD simulation and you want to separate molecules into HS around specified molecule, who guaranties that the thermodynamic criteria (like minimum on g(r)) is applicable to your instant configuration? Maybe a molecule occasionally occurred before the boundary for the FHS, but it will move back to second hydration shell in the next moment. This question arise in the calculation of residence time.

The structure of hydration shell is very different for anions and cations (see Fig.2).

Around cation the waters are oriented in such a way, that their dipole moments lie in the same line with the cation charge. The negative charged oxygens are closer to ion in this case. Around anion the waters are oriented in a way, that one oxygen-hydrogen bond lies in the same line with the cation, and the dipole moment is biased from this line. These differences result in asymmetry of //cation and anion hydration//: thermodynamics of hydration for cations and anions is different.

Coordination number
> **Coordination number - average number of water molecules in the first hydration shell of solute molecule.** Simulation community uses this definition.

But the term //coordination number// has different meaning: 1. In [|coordination chemistry] it means the number of ligands, that are bounded ("coordinated") by "donor-acceptor" mechanism to a coordinating ion, molecule... 2. The number of neighboring particles in the direct contact to the given particle in crystal.

More information about these definitions see: []

The //coordination number// also can be defined as a average number of water molecules in the whole hydration shell of solute.

The coordination number can be calculated from the radial distribution function in the case of homogeneous liquid. The running coordination number is: > math n^{\alpha}_{\beta}(r) = 4\pi \rho_{\beta} \int_0^\infty g_{\alpha \beta}(r) r^2 dr math where math n^{\alpha}_{\beta}(r) math - the running coordination number (average number of particles //\beta// in the sphere with radius //r// around particle //\alpha//, math g_{\alpha \beta}(r) math - //\alpha-\beta// radial distribution function, math \rho_{\beta} math - number density of particles //\beta// in the system.
 * How to calculate:**

The coordination number of particles //\beta// in the first solvation shell of particles //\alpha// is just the value of running coordination number function at the //r// = the position of the first minimum on the g(r) function. (See the criteria for 1st HS in Hydration shell).

Note, that when the solvent is water the //coordination number// is also sometimes called **hydration number**.

Residence time
(See also []). Residence time of water molecule in **the first hydration shell** (**FHS**) is the mean time one water molecule stays within the first hydration shell. The escape of water molecule from the FHS can be regarded as the reaction process: > math ion-water_n = ion-water_{n-1} + water^{bulk} math where n - coordination number.

Thus, from the kinetic theory we have the relation between residence time and the energetic barrier along the reaction coordinate between initial (ion-water_n) and final states (ion-water_{n-1}): > math \dfrac{1}{t_{R}} \propto e^{-\beta E_a} math where math t_{R} math - residence time, math E_a math - [|activation energy] of the process (energetic barrier along the reaction coordinate).

//How to calculate:// The residence time is found through fitting the auto correlation function of water molecule escape: > math R(t) = \frac{1}{N_h} \sum_{i=1}^{N_h} \left\langle \theta_i(0) \cdot \theta_i (t) \right\rangle math with exponent decay function: > math exp(-t/t_R) math where: math \theta_i (t) math - the Heaviside step function, equal to //1// if the water molecule //i// is in the first coordination shell of the ion at time //t// and to zero otherwise.
 * 1).** The conventional way to calculate the residence time is the method proposed by Impey et al. (Ref.1).

Additional criteria, absence time: > math t_{absence} math - if a water molecule, that left the hydration shell, returns back in time less then the absence time then it is considered being not left the hydration shell.

Imprey proposed the absence equals to 2 ps without clear and rigorous explanation why. Most of the authors (Rasaiah, Lynden-Bell ...) usually provide calculations with t_{absence}=0 ps and t_{absence}=2 ps.

An example of calculation with the method 2 could be found in Ref.2.
 * 2).** Another way to calculate residence time is to just use the definition: count the average time water molecule stays within the FHS. This method is found to be very sensitive to the input parameters - boundary of the FHS, absence time ... And the residence times calculated with this method are very distinct from those calculated with method 1: they are at least 10 times smaller.

> //References:// 1. Impey, Madden, McDonald //J.Phys.Chem.// **1983**, //87//,5071. 2. Frolov, Rozhin, Fedorov //Chem.Phys.Chem.// **2010**, submitted

Samoilov energy
The free energy profile along the reaction coordinate of the water escape from the FHS (see Residence time) is given by the ion-water Potential of Mean Force. Thus the [|activation energy] is the difference of the first maximum and the first minimum on the ion-water PMF (PMF has to be slightly modified because of spherical coordinate system, see Ref1.): > math E_a = PMF_{ion-water}(r_{1 max}) - PMF_{ion-water}(r_{1 min}) math

The same is valid for the water molecule escape from the FHS of water molecule: > math E_a^0 = PMF_{water-water}(r_{1 max}) - PMF_{water-water}(r_{1 min}) math

The residence time of water molecules is proportional to the activation energy (see Residence time). The relative residence of water in FHS of ion and in FHS of water should be proportional to the difference in the activation energies of these two processes (Ref.2,3). Finally we get: > math \dfrac{t_{R}^{sol}}{t_{R}^{0}} = e^{\beta (E_{a}-E_a^0)} = e^{\beta \Delta E_{Samoilov} math where math t_{R}^{sol} ~ and ~ t_{R}^{0} math - the residence time of water molecule in the HS of solvent and the residence time of water molecule in the HS of bulk water molecule. math \Delta E_{Samoilov}=E_{a}-E_a^0 math - **Samoilov energy**(Ref.3). It shows the difference between the energy of activation of water molecule release from HS of an ion and a water molecule in bulk.

//References:// 1. Koneshan, Rasaiah et al. //J.Phys.Chem.B// **1998**,//102//,4193. 2. Chong and Hirata //J.Phys.Chem.B// **1997**,//101//,3209. 3. Samoilov //Discuss.Faraday Soc.// **1957**,//24//,141.

Positive and negative hydration
According to Samoilov (Ref.1), the //hydration// is called //positive//(//negative//) if the residence time of water molecule in hydration shell of solute is //bigger//(//smaller//) then the residence time of water in hydration shell of water molecule in the bulk solvent.

In mathematical way it reads: > math \begin{cases} t_{R}^{sol} \geq t_{R}^{0} ~ and ~ \Delta E_{Samoilov} \geq 0 &\text{- positive hydration,}\\ t_{R}^{sol} < t_{R}^{0} ~ and ~ \Delta E_{Samoilov} < 0 &\text{- negative hydration.} \end{cases} math where math t_{R}^{sol} ~ and ~ t_{R}^{0} math - the residence time of water molecule in the HS of solvent and the residence time of water molecule in the HS of bulk water molecule.

Or alternatively, in terms of Samoilov energies: > math \begin{cases} \Delta E_{Samoilov} \geq 0 &\text{- positive hydration,}\\ \Delta E_{Samoilov} < 0 &\text{- negative hydration.} \end{cases} math

//References:// 1. Samoilov //Discuss.Faraday Soc.// **1957**,//24//,141. 2. Chong and Hirata //J.Phys.Chem.B// **1997**,//101//,3209.

Kosmotropes and Chaotropes
([] []) > ** Kosmotrope (structure-maker) and chaotrope (structure-breaker) - a substance that increase and decrease the structuring of solvent around it comparing to pure solvent. We will use this notation.** The terms is very general and not strict. Very often they are used to indicate the ability of substance to stabilize or destabilize the protein structure in solution (Ref.1, see Wiki). In this case, these terms are strongly related to Hofmeister series, and indicate where the substance stays in these series.

According to our definition, we can distinguish kosmotropes and chaotropes by their effect on the entropy of water. Imagine we are looking at water molecules in the close vicinity of solute: if the water is more structured, less "free" then in bulk => then the entropy of water is smaller then in bulk, and the solute is called //kosmotrope//. If water in the vicinity of solvent is more mobile then in bulk, then it is called //chaotrope//.

On the Figure 1 you can see how it is possible to separate alkali and halide ions to //kosmotropes// and //chaotropes//, basing on the entropy criterion.



Alkali and halide ions aligned according to strength of ion-water interactions (Ref.2): > math kosmotrope ~ ~ ~ ~ chaotrope math > math Li^+,~ Na^+ ~ || ~ K^+,Rb^+,Cs^+ math > math \: \: F^- ~ || ~ Cl^-,Br^-,I^- math where || symbol shows the strength of water-water interactions.

You see that strongly hydrated (small size, high surface charge) ions are //kosmotropes// and the weakly hydrated (big size, low surface charge) ions are //chaotropes//.

Please, remember that the definitions //kosmo-// and //chaotropes// are not strict and may vary!.
 * Note**

//References:// 1. Marcus //Chem. Rev.// **2009**,//109//,1346. 2. Collins, Neilson and Enderby //Biophysical Chemistry// **2007**,//128//,95.

Collins pyramid
Collins "volcano" plots illustrate the "Law of matching affinities". It states that if cation and anion have similar affinities (strength of interaction) to water molecules, then they easily form contact pairs in solution; but if they have big difference in their affinities, then will not form ion pair.

Possibility to form ion pair is very important since it relates to the solubility of salt, entalphy of dissolution and so on.

> **Heat of dissolution - heat that releases in the process of transfer of one ion pair from crystalline phase into solution (at constant temperature and pressure).**

> **Enthalpy of dissolution - heat that absorbs in the process of transfer of one ion pair from crystalline phase into solution at constant temperature and pressure.**

> math \Delta Q_{dissolv} = - \Delta H_{dissolv} math

//Explanation:// The explanation of "Law of matching affinities" is hidden in the interplay of strength of the ion-ion, ion-water and water-water interactions. Let us consider small highly hydrated ions like Li+ and F-. The interaction between these ions is much stronger then their interaction to water molecule and they tend to stay together. Thus, separation of the ions in solution will require additional energy, that is why the enthalpy of dissolution of LiF is positive (the heat is absorbed). (see Fig.2) Let us consider big weekly hydrated ions like Cs+ and I-. The interaction of water with itself is stronger then the ion-water interactions and even more stronger then cation-anion interactions. Thus these ions "want" to be hydrated (thus separated in solution), but water "prefers" to interact with water not with ions. As the result, water respells the ions and makes them stay together and form ion pairs. The enthalpy of dissolution of CsI is positive (the heat is absorbed). In the cases of ions with very different affinities to water (f.e. Li+ and I-, Cs+ and F-) the situation is different. The formation of a contact pair in water leads to partial dehydration of ions. It turns out that the energy of partial dehydration of strongly hydrated ion costs more, then the energy gain in formation of ion pair with weakly hydrated ion with opposite charge. The enthalpy of dissolution becomes negative (the heat is released).



//References:// Collins, Neilson and Enderby //Biophysical Chemistry// **2007**,//128//,95.

Hofmeister series
(See also [], [])

The Hofmeister series align the salts according to their ability to precipitate a given protein from aqueous solution (Ref.1). It is found that the ability of the salt to precipitate a protein is determined by the properties of anion. For a given cation, the anions sorted according to their precipitation ability like: > math CO_3^{2-} > SO_4^{2-} > F^- > Cl^- > Br^- > I^- > SCN^- math Bigger the surface charge of anion - more effective the anion for precipitation.

Series for cations are also exist but they are less developed: > math (CH_3)_4N^+ > K^+ > Na^+ > Cs^+ > Li^+ > NH_4^+ > Ca^{2+} math Here, the relation between the surface surface charge of ion and its precipitation ability is not obvious.

The mechanism of Hofmeister series is not completely understood. Since this effect is complicated, usually the result of different effects working together: increase of the surface tension by salts, selective binding of ions to protein groups, concurrence between protein groups and ions for the water molecules, occurrence of salt bridges and so on. The most complication is that all these effects are effects on a "surface" which are far from being completely understood nowadays.

//References:// Section 6.1 in review: Marcus //Chem. Rev.// **2009**,//109//,1346.

"Salting in" and "Salting out"
See [] and []

//"Salting out"// has two meaning: an effect and a method.

//"Salting out"// (//"Salting in"//) is the decrease (increase) of solubility of dissolved substance (f.e. protein) by increasing the  concentration of particular salt.
 * An effect:**

F.e. some of ions (and consequently salts) from Hofmeister series can decrease the cancentration of proteins - this is //salting out effect//. Rarely, but possible that ions increase the solubility of protein - //"salting in" effect//.

//Salting out// - is the method to separate different substances exploiting the fact that the solubility of the substances is differently influenced by increase of salt concentration. See [].
 * A method**

The mechanism of salting out/in is rather complex for proteins and other bio-polymers. But, let us consider a purely hydrophobic solute (f.e. carbon nanotube CNT). Addition of salt (with highly hydrated ions) into the solution will result in "Salting out" of carbon nanotubes. In this case the whole effect might be explained by the increase of surface tension of solution. The CNT has a surface thus the solvent has to form a "surface" around CNT. Highly hydrated ions increase the surface tension of solution (Ref.1). Thus an addition of the salt will force the solution to decrease any surface as much as possible. And hence force the CNT go out solution -> decrease its solubility.

//References:// 1.Collins, Washabaugh //Q.Rev.Biophys.// **1985**,//18//,323.

Surface tension
> **Surface tension - reversible work for creation of a unit square of the surface in the system.**

> math \sigma = \left( \dfrac{\partial G}{\partial A} \right)_{T,P,n} math where //G// - Gibbs free energy

Solvation Free Energy
[] > "**Solvation energy** = the change in Gibbs energy when an ion or molecule is transferred from a vacuum (or the gas phase) to a solvent. The main contributions to the solvation energy come from: (a) the cavitation energy of formation of the hole which preserves the dissolved species in the solvent; (b) the orientation energy of partial orientation of the dipoles; (c) the isotropic interaction energy of electrostatic and dispersion origin; and (d) the anisotropic energy of specific interactions, e.g. hydrogen bonds, donor-acceptor interactions etc." (citation from the given link:IUPAC Compendium of Chemical Terminology 2006)

Debye and Bjerrum length
See (highly recomended): [] []

Both parameters ("length") describe how efficiently the coulomb interactions are screened in solution. Imagine you have two ions in vacuum. The interaction between the two ions is just the coulomb potential. Now consider the two ions in water salt solution. There are many water molecules and ions in between these two ions. The effective potential between two ions is now given by Potential of Mean Force. The water molecules and other ions between the two ions possess dipoles and charges. But there equilibrium distribution around ions will decrease the effective potential between the two ions.

> **Bjerrum length - "the separation at which the electrostatic interaction between ions in solution is comparable in magnitude to the thermal energy scale, kBT"**. where kB is the Boltzmann constant and T is the absolute temperature.

The formulae explanation of Bjerrum length: > math U_{ion-ion}(r=\lambda_B) = 1 k_BT math where U - screened coulomb potential between two ions in solution. > **Debye length - parameter in the exponential scaling factor to coulomb potential**. (Maybe not rigorous) See the Debye-Hueckel theory to understand it more.

The screened coulomb potential between two ions having charges (//q_i// and //q_j//) in solution is written as: > math U_{ion-ion}(r>>\sigma) = \dfrac{q_iq_j}{r}e^{-r/\lambda_D} math where math \lambda_D math - Debye screening length. The formula is valid only for sufficient separations between ions (r>>\sigma).

Sometimes Debye length is denoted as: > math \lambda_D = k^{-1} math

The connection between these two screening coefficients see in [].

Crystal radii database
[]

Water models
[]

=SaltCrashCourse_TASKS=

List of tasks:
Example: property = size ---      F-    Cl-   Br-   I- size:  small   >  big
 * 1.** For the line of ions write the matching properties on the left and right:

---                       F-      Cl-     Br-     I- size: surface charge: chao/kosmotrope: Strength of hydration: Positive/negative hydr.: Value of hydration number: Value of the 1st min on PMF*:
 * Fill the table:

\* PMF = ion-water PMF as a function of the distace r between ion and water.


 * The same for Li+ Na+ K+ Rb+ Cs+


 * 2.** Explain term "asymmetry of cation and anion hydration". Which ions are strongly hydrated cation or anions?.

Plot the Collins volcano plots for them. Mark the areas where ions tend to form direct contacts and where they want to stay separately in solution.
 * 3.** Form all possible salts out of Li+ Cs+ F- I- ions.

F-   I-   Br-   Cl-. Explain your choice.
 * 4.** Ion interaction with solutes:
 * Align the anions according to their "salting out" ability of a hydrophobic solute for an arbitrary cation:

Explain your choice. (which effects make the "Hofmeister effect" more complicated comparing to just a hydrophobic solute? List at least 3)
 * Align the anions into Hofmeister series for arbitrary cation:   F-    I-   Br-   Cl-.


 * 5.** Calculate from the given ion-water radial distribution function:
 * determine the boundaries of the 1st and 2nd hydration shells

and the coordination numbers in the first and second hydration shells. NOTE!: do not spend more that 5 min on this task! You can just explain how to do this.
 * calculate the running coordination number (number density = 1 mol/sig^3)

(residence time for "water around water" = 1 ps   residence time for "water around ion" = 1 ps) Which kind of hydration is it (positive/negative)?
 * calculate the Samoilov energy
 * calculate the activation energy of water molecule release out of the 1HS of water having

g(r): (r is given in \sigma diameters of water molecule) 0        0    0.5000         0    1.0000    0.0100    1.5000    2.0000    2.0000    0.5000    2.5000    1.5000    3.0000    1.0000