What is “chemical pressure”?

Question

While reading some publications on solid superconductors, I encountered a term "chemical pressure" a few times, which is usually attributed to the changes of superconducting transition temperature $T_\mathrm{c}$ alongside with the hydrostatic, or physical, pressure. For the reference, consider this paper: Chemical Pressure and Physical Pressure in $\ce{BaFe2(As_{1-x}P_{x})2}$ [1].

Whereas I can understand what the physical pressure is and how it's defined and estimated, I found no sources which would explicitly define either what chemical pressure is, or how it's measured (and what units are used) and/or analyzed.

All explanations are somewhat expansive and usually point to the effects of isomorphic substitution in the crystal lattice (ionic structure), or intermolecular repulsion between ligands (molecular structure). So, what chemical pressure is in the nutshell, how it's measured and expressed?

Reference

1. Klintberg, L. E.; Goh, S. K.; Kasahara, S.; Nakai, Y.; Ishida, K.; Sutherland, M.; Shibauchi, T.; Matsuda, Y.; Terashima, T. Journal of the Physical Society of Japan 2010, 79 (12), 123706 DOI: 10.1143/JPSJ.79.123706. (Open Access)

Answer 1

External and internal pressure

To study the effect of pressure on properties of a solid, is equivalently to learn how changes in volume transform physical parameters. For external pressure at constant temperature, this relationship manifests through compressibility $\kappa$.

$$\kappa =-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$$

An alteration in volume will modify the energy of a system. The relationship is internal pressure, something chemists first encounter when discussing real gases.

$$\pi_T=\left(\frac{\partial U}{\partial V}\right)_T$$

In solids, one might discover different notations. For example^[1],

$$\alpha =\left(\frac{\partial W}{\partial V}\right)_T \sim \frac{\mathrm{d}W}{\mathrm{d}V} $$

where $W$ is potential energy, and $\alpha $ is called energy–volume correlation. Disregarding possible sorption, varying external pressure is the purest option of probing internal pressure.^[2] Unfortunately, this might not always be feasible.^[2][3] Most actinides, for instance, are highly radioactive, thus limiting their study to very specialised laboratories.^[3] Many actinides are too scarce to perform necessary large scale measurements.^[3] Here is where chemical pressure comes in.

Chemical pressure

Instead of external pressure or thermodynamic internal pressure, we now discuss chemical pressure. Much akin to external pressure, scientists take interest in how energy is dependent on volume. Chemical pressure comprises of substituting in or adding chemical entities to a solid network.^{[2] [4] [5] [6]}

• External pressure presses down on the solid. Volume is modified via a compressibility relationship.^[1] A more direct or pure method.^[2]
• Chemical pressure is induced by changing cell volumes via the introducing of new chemical entities, such as atoms or compounds, in the lattice. An indirect, less pure approach.^[2]

Obviously, chemical doping modifies a system in other ways besides volume alone.^{[2] [5] [7]} By choosing an element with similar electronic configuration, these deviations from ideality are minimised.^{[2] [7]} Another advantage of chemical pressure over extrernal pressure is that the former may be positive and negative.^{[2] [8]} Positive pressure implies that pressure from chemical substitution is directed inwards, and adds to external pressure. Negative pressure pushes the solid apart, hence acting against external pressure.^[2]

Immunisation against terminological confusion

Condensed matter science, like other fields of study, is entitled to its own nomenclature. For chemists specialising elsewhere, this might seem a superfluous obfuscation. This is especially true for cutting-edge chemistry where terminology is fresh and perhaps not entirely settled. For this reason, synonyms are widespread, and overlap with other terms may occur.

• External pressure is also called physical pressure^[4] to establish a physical pressure – chemical pressure dichotomy.
• Chemical pressure is often enclosed between quotation marks. More confusingly, internal pressure is used as an alternative to chemical pressure.^[5] Perhaps this is to emphasise an external pressure – internal pressure dichotomy. Or, it could be, that since chemical pressure is used as a tool to measure internal pressure, the concepts have blurred lines because the distinction seemed unnecessary.

Units and experiments

Internal chemical pressure will probably have concurring units with external physical pressure. Pascals ($\ce{Pa}$) when operating in the SI framework. Very few studies explicitly mention this fact^[5] because more often the changes in some concrete parameter are discussed^{[1] [9]}. Because of what chemical pressure constitutes, we do not apply a chemical pressure of $10\ \mathrm{kPa}$. We might, however, bring about (or apply) a chemical difference in pressure ... equivalent to several $\mathrm{kbar}$ of external pressure^[5]. For solids where external pressure experiments are currently unthinkable, this equivalence cannot be discussed. But the goal is not to compare chemical pressure to external pressure anyway; as before, we wish to learn more about how volume affects (potential) energy.

As far as I am aware, chemical pressure cannot be measured directly. Chemical pressure is induced inside a solid material by substituting or introducing new chemical entities. Where would one put the barometer? If it is possible to carry out external pressure measurements alongside chemical doping, we may compare one to the other. To emphasise, the effects need not match. In these and other cases, volume changes in cells are estimated or measured, and their effect on the system as well.

/(short overview/list of experiments in progress, next week)/

In my opinion, this subsection deserves a question on its own.

TL; DR

/(in progress, next week)/

Additional reading

For an overview of highly-correlated electron systems in metallic solids, I recommend C. Marini. Pressure-induced metallization process in Strongly Correlated Electron Systems [Online]; G. Stefani, G. Altarelli, P. Postorino, Eds. Università degli Studi Roma TRE: Rome, Italy, 2009–2010. Archived link: bit.ly/2yPJcCo

References and bibliography

• [1] G. Borelius. 'Internal Pressure in Solids and Liquids'. Physica Scripta, 1970, 1, 2-3, pp 141–147. DOI: 10.1088/0031-8949/1/2-3/011
• [2] J-M. Fournier. 'Chemical pressure in actinide systems'. Physica B: Condensed Matter, 1993, 190, 1, pp 50–54. DOI: 10.1016/0921-4526(93)90441-8
• [3] U. Benedict. 'Properties of Actinide Metals Under High Pressure'. Journal de Physique Colloques, 1984, 45, C8, pp C8-145–C8-148. DOI: 10.1051/jphyscol:1984826
• [4] D. M. Friedman. New Research on YBCO Superconductors; Y. Itoh, Ed. Nova Science Publishers: 2008. Chapter 1: Recent Advances of NMR and NQR Studies of $\ce{YBa2Cu3O_{7-δ}}$ and $\ce{YBa2Cu4O8}$, pp 25–69 Subsection 6: 'Chemical Pressure, Physical Pressure, and Site Disorder', pp 59–59. ISBN: 978-1-60456-084-8
• [5] A. Hauser, A. Nahid, S. Delahaye, A. Sadki, S. Schenker, R. Sieber, M. Zerara. 'Chemical Pressure'. CHIMIA International Journal for Chemistry, 2002, 56, 12, pp 685–689. DOI: 10.2533/000942902777679858
• [6] R. S. Liu, C. H. Shen, T. S. Chan, R. Gundakaram, S. F. Hu, J. G. Lin, C. Y. Huang. 'Chemical Pressure Induced Phase Transition in Single and Double Perovskites with Magnetoresistance Effect'. Tamkang Journal of Science and Engineering, 2002, 5, 1, pp 59–61. Archived link: bit.ly/2fvxrsv
• [7] A. Huon, D. Lee, A. Herklotz, M. R. Fitzsimmons, H. N. Lee, S. J. May. 'Effect of chemical pressure on the electronic phase transition in $\ce{Ca_{1-x}Sr_xMn7O12}$ films'. APL Materials, 2017, 096105, pp 096105-1–096105-7. DOI: 10.1063/1.4994089
• [8] Masatomo Uehara, Tsuyoshi Amano, Sachiko Takano, Tatsuya K[o]ri, Takahiro Yamazaki, Yoshihide Kimishima. 'Chemical pressure effect on the superconductor $\ce{MgCNi3}$'. Physica C, 2006, 440, 1–2, pp 6–9. DOI: 10.1016/j.physc.2006.03.017
• [9] M.Tropeano, C.Fanciulli, F.Canepa, M.R.Cimberle, C.Ferdeghini, G.Lamura, A.Martinelli, M.Putti, M.Vignolo, A.Palenzona. 'Effect of the chemical pressure on superconductivity and SDW in undoped and $15$% $\ce{F}$ doped $\ce{La_{1-y}Y_yFeAsO}$ compounds'. Physical Review B, 2009, 79, 174523, pp 174523-1–174523-6. DOI: 10.1103/PhysRevB.79.174523

Answer 2

First off, "chemical pressure" (CP) is often portrayed as an empirical concept or effect for linking chemical composition and physical properties in the first place, and not a physical quantity -- even though it is one. That might be a reason for certain confusion.

Doing some research I encountered "CP" term being used in other fields with substantially different meanings:

• Chemical engineering: a pressure created by the reactants inside an apparatus;
• Medicine: a negative stimuli of narcotic drugs; also effect of hormones on sexual behaviour;
• Biology: a synonym for osmotic pressure;
• Ecology: a suppression caused by prolonged exposure to insecticides;
• Industry: "Chemical Pressure Company, Inc." (USA) manufactured polymers in 1970-80s.

One of the first reliable literature sources where "CP" has been coined as a truly chemical term used these days, was an exceptionally well-written review "A brief review of a study of cohesion and chemical attraction" [1]. To the mid-1920s van der Waals concept of "incompressible" atoms has been criticized, and transitory collisions in gases were compared to the "united" atoms in solids and liquids. Here are the key points (emphasis mine):

When a molecule is composed of more than one atom, different intrinsic internal pressures must evidently exist on different sides of each atom ... For instance in liquid bromine ...,evidently the two atoms in a molecule are bound together much more firmly than in the mere cohesion of the liquid. Hence much greater pressure (and therefore much greater compression) must exist between the atoms of a given molecule than exists between two different molecules. The first-named pressure, the greater one, may be called chemical intrinsic pressure; the second, the smaller one, may be called cohesive intrinsic pressure. They are both probably due in this case (as in others) to the same forces being exerted to different extents. In such cases the atomic fields of force must be distorted, the atomic structure being far more compressed on one side than on all the other sides. [...]

Evidently the main influences which work in any physicochemical atomic contacts may be represented by the following equation of two terms indicating compressing agencies, with two terms indicating distending agencies: $$p + \Pi = \Pi_p + P_\Theta \tag{1}$$ in which $p$ represents external pressure, $\Pi$ the sum of all possible intrinsic compressing effects; $\Pi_p$ the intrinsic distending or "repulsive" pressure, and $P_\Theta$, thermal pressure. This equation is expressed as between pressures rather than forces because the first and the last terms may then be directly measured; for $p$ may be found by an ordinary pressure gauge, and $P_\Theta$ equals $T_\alpha / \beta$ where $\alpha$ and $\beta$ are coefficients of expansion and compression respectively. [...]

Common observation of the phenomena of cohesion and chemical combination shows that both $\Pi$ and $\Pi_p$, must decrease as the distance of the particles (or the volume of the system) increases. Thus we may write for a solid or liquid with monatomic molecules: $$p + \Pi_0 \left( \frac{v_0}{v} \right)^m = \Pi_p \left( \frac{v_0}{v} \right)^n + T_\alpha / \beta \tag{2}$$ while admitting that the exponents $m$ and $n$ are not necessarily constant. $n$ must be larger than $m$, otherwise the system could not adjust itself to equilibrium, and a slight increase of external pressure would cause the solid or liquid to collapse. [...]

Heats of evaporation indicate that the exponent m is probably not far from 2, as van der Waals assumed, and the other exponent ($n$) may likewise be evaluated (in a somewhat complicated way) with reasonable probability. [...]

... chemical combination, in polyatomic elements or compounds, involves much more complicated treatment. ... the compressibility of a binary compound may be roughly pictured by the following equation, in which $x$ is the fraction of each atom subjected only to cohesive pressure $\Pi_1$ and $(1 - x)$ is the fraction subjected to the intense chemical pressure $\Pi_2$. $$\beta \approx \frac{x}{n_1 - m} \left( \frac{x}{\Pi_1} + \frac{x-1}{\Pi_2} \right) \tag{3}$$

... the importance of the idea which postulates the existence of different pressures on different sides of a single atom is of great importance. ... whereas in ethyl ether the cohesive pressure is probably of the order of 2000 atmospheres, the intrinsic chemical pressure existing between atoms of carbon and hydrogen or carbon and oxygen (different in each case) must be much over 100,000 atmospheres. Precisely the same sort of difference, although perhaps of less degree, must exist within anistropic crystals, even when they are built up of only one kind of atom (e.g., zinc). [...]

... every substance must exist as such by means of a balance of pressures: external pressure plus affinity pressure equals intrinsic distending pressure plus thermal pressure.
Chemical affinity pressure must differ from cohesive pressure mainly in its intensity and in the fact that the chemical union of two atoms must involve great atomic distortion on one side of each, not equalled by the effect of the cohesive pressure around the periphery of the molecule.
Simple cases involving the above considerations have been subjected to mathematical analysis, with reasonably satisfactory results. Very high internal pressures in non-volatile solids are indicated, having the order of the breaking strength of very rigid substances like glass. These pressures are consistent with the heats of chemical combination and the heats of evaporation.

Important conclusion is also that CP is defined as an actual physical quantity and should be treated accordingly. So, the same pressure units (Pa, bar, atm etc.) are applicable. Also, elemental substitutions create local stresses in a crystal structure and therefore might substantially affect CP.

The next milestone probably was a short review from late 1960s "Das „Druck-Abstands-Paradoxon”︁" [2] (English "The "pressure-distance paradox""), which attributed chemical pressure as a counterpart to the external physical pressure, influencing interatomic distances, coordination numbers, phase transformations, densities and unit cell dimensions in various systems: graphite - diamond, $\ce{BN}$, $\ce{SiO2}$, $\ce{ZnS}$, $\ce{ZnSe}$ and $\ce{AgI}$.

Since 1980s CP is widely used as a concept for the description of chemical substitution and phase transition in solids. It was shown that chemical pressure coefficient ($\epsilon$, [Pa]) can and should be used to discover the symmetry of the solid phase and dilution coefficient (viz. $x$ from eq. 3, here denoted as $t$) by applying the counterbalance condition $P + \epsilon t = 0$ ($P$ is physical pressure) to the phase diagram [3] (emphasis mine):

The formulation of guide-lines for the chemical pressure ($P^*$) concept represents a difficult task. Generally chemical pressure manifests itself as any response of the crystal lattice to a given change in substitution. A reasonable requirement is that all qualitative features of the pressure, temperature ($P, \mathrm{T}$) phase diagram for a given solid solvent ($T$) should be retained in the composition, temperature ($t, \mathrm{T}$) phase diagram for its solid solution systems, and that only the phase boundaries are shifted by the substitution. It is convenient to consider the $P, \mathrm{T}$ phase diagrams for different substitution levels ($t$) of a solid solution phase as made up of (roughly) additive contributions from a chemical component (which can be positive or negative) and an external (positive) pressure. ... The shift of the $P, \mathrm{T}$ phase boundaries as a function of $t$ can be used as a quantitative measure of the chemical pressure.

Interestingly, from the thermodynamics point of view, chemical potential $\mu_i$ is defined as a subset of CP [4, p. 206]; so CP can also be quantitatively expressed in units of heat/energy.

During the past decade several advancements in simulation of CP by quantum mechanical approach (DFT-CP, Hirshfeld surface analysis) [5] and EXAFS analysis [6] have been made.

References:

1. Richards, T. W. Transactions of the Faraday Society 1928, 24, 111 DOI: 10.1039/tf9282400111.
2. Kleber, W. Krist. Techn. 1967, 2 (1), 13–14 DOI: 10.1002/crat.19670020103. (in German)
3. Ziȩba, A.; Zach, R.; Fjellvåg, H.; Kjekshus, A. Journal of Physics and Chemistry of Solids 1987, 48 (1), 79–89 DOI: 10.1016/0022-3697(87)90144-2.
4. Weinhold, F. Classical and Geometrical Theory of Chemical and Phase Thermodynamics; Wiley: Hoboken, N.J, 2009. ISBN 978-0-470-40236-8.
5. Berns, V. M.; Engelkemier, J.; Guo, Y.; Kilduff, B. J.; Fredrickson, D. C. J. Chem. Theory Comput. 2014, 10 (8), 3380–3392 DOI: 10.1021/ct500246b.
6. Mukherjee, S.; Nag, A.; Kocevski, V.; Santra, P. K.; Balasubramanian, M.; Chattopadhyay, S.; Shibata, T.; Schaefers, F.; Rusz, J.; Gerard, C.; Eriksson, O.; Segre, C. U.; Sarma, D. D. Physical Review B 2014, 89 (22) DOI: 10.1103/PhysRevB.89.224105.