Tangential electrostatic field at metal surfaces

2.1.1 Electrostatic equilibrium conditions (EECs) for metals

For a metal placed in a complex electrostatic environment, the surrounding charges and electric field may exert an external electrostatic potential on the metal. The application of the external potential does not usually influence the Fermi level and the intrinsic electrostatic potential φ _i(r) of the metal. Considering the intrinsic potential φ _i(r) and the external potential, the total electrostatic potential of the metal may be written as the following equation, which is a result of the superposition principle for the electric fields and electrostatic potentials [18,19,20]:

where φ(r) is the total electrostatic potential for the metal and it is usually position dependent, and φ _e(r) denotes the external electrostatic potential including the contribution from the induction charges at the metal surface. Correspondingly, the electrostatic field inside the metal may be given as follows:

where E(r) is the total electrostatic field in the metal and it is given by E(r) = −∇φ(r), E _i(r) is the intrinsic electrostatic field and it is given by E _i(r) = −∇φ _i(r), E _e(r) is the electrostatic field induced by the external potential and it is given by E _e(r) = −∇φ _e(r). Upon reaching the electrostatic equilibrium, very interestingly, the external electrostatic potential φ _e(r) is position independent and is a constant for the whole metal according to Yuheng Zhang equation. So, it may be concluded that a metal is an equipotential body where the “potential” refers to the external electrostatic potential. In other words, the electrostatic field E _e(r) induced by any external electrostatic potential φ _e(r) is always zero inside the metal, i.e.,

Meanwhile, the external potential-induced electrostatic field at the metal surface must be normal to the surface and the related TEF must be zero at the metal surface,

where E _et(r) stands for the TEF component of E _e(r) at the metal surface. Thus, the traditional EEC for a metal should be modified and summarized as follows:

1. A metal cannot be an equipotential body, but it must be an “equipotential” body where the “equipotential” refers to the external electrostatic potential containing the contribution of induction charges at the metal surfaces.

2. Inside a metal, an intrinsic electrostatic field always exists, but the electrostatic field generated by any external electrostatic potential must be zero.

3. An intrinsic TEF is popular at the metal surface, but the TEF originating from any external electrostatic potential must be zero at the metal surfaces.

4. The intrinsic net charges due to the intrinsic electrostatic field can exist inside a metal, but the externally charged or discharged charges cannot reside inside the metal and must be distributed at the metal surfaces.

In the above statements on the modified electrostatic equilibrium conditions (MEEC), a metal refers to any real single-component metal or multi-component alloy which contains many “freely” moving electrons and exhibits a large electrical conductivity. Different from definitions in many textbooks [4,18,19,20], it does not require a uniform composition and an infinite electrical conductivity. The MEEC may be appropriate under the conditions that the state of the metal, e.g., stress distribution, the number and distribution of various defects, and composition distribution, keeps invariant on the application of external electrostatic potential. Nevertheless, in some extreme cases, the externally applied electrostatic potential can change the state and indirectly induce an electrostatic field inside the metal, which may thereby affect the electrostatic potential and the distribution of charges in the metal.

Comparing with the EEC, the MEEC may exhibit some advantages. First, the MEEC may not be restricted to a perfect conductor, and could be valid for any real metal or alloy with noticeable electrical conductivity regardless of the state. So, the MEEC may have a larger application scope than EEC. Second, the physics revealed by the MEEC may be more correct and more appropriate. Third, the MEEC could point out the popular existence of the intrinsic electrostatic field inside a metal and the TEF at the meal surface, which may assist in interpreting the related phenomena in various fields.

2.1.2 Electrostatic boundary conditions (EBCs) for a metal

To solve the electrostatic problems of metals, one usually makes use of the traditional EBC which is given by [1,4,18,19]

where n denotes the unit vector perpendicular to the metal surface, pointing outwards, E is the electrostatic field at metal surface, σ is the surface charge density, and ε ₀ and ε are the vacuum permittivity and relative dielectric constant of the media around the metal, respectively. Nevertheless, both the intrinsic TEF at the metal surface and the intrinsic electrostatic field inside metals [9] may indicate that the EBC may not be applicable for any real metal. So, the rigorous electrostatic boundary conditions (REBC) for a metal should be tackled and they are given by

where ε _m is the electrostatic dielectric constant for the metal, E is the total electrostatic field outside the metal surface, and E _i(r)∣ _S−0 is the intrinsic electrostatic field under the metal surface. As is shown, the TEF at the metal surface is continuous. The electric field outward normal to the metal surface cannot precisely determine the surface charge density and it is also dependent on the intrinsic electrostatic field inward normal to the metal surface. It may be easily found that only when the intrinsic electrostatic field E _i(r) is so weak that it can be ignored, can the REBC be simplified to be the EBC.

For the REBC, the surface charge density may be written as σ = σ _e + σ _i, where σ _e is the externally charged or discharged charge density at the metal surface, and σ _i denotes the intrinsic surface charge density which only depends on the intrinsic electrostatic field distribution, and it may be expressed as follows:

where E _i (r)∣ _S+0 is the intrinsic electrostatic field above the metal surface. Substituting Eqs. (6), (8), and (11) in Eq. (10), the REBC may be expressed in another form

The form of REBC gives the EBC for the electrostatic field E _e(r). A comparison between EBC (Eq. (9)) and REBC (Eq. (12)) may be interesting and enlightening. At first glance, their forms are the same as each other, but their physical meanings may be quite different from each other. EBC demonstrates that the electrostatic field at the metal surface must be normal to the surface, and its magnitude determines the surface charge density. On the contrary, REBC indicates that the externally applied electrostatic field E _e(r) is perpendicular to the metal surface and the electrostatic field E _e(r) at the metal surface is related to the externally charged or discharged surface charge density σ _e.

The REBC not only can lead to complexities and difficulties for solving the electrostatic problems of metals, but can also lead to new understandings of electrostatics of metals.

2.1.3 First uniqueness theorem for metals

The first uniqueness theorem states that “The solution to Laplace’s equation in some volume V is uniquely determined if the potential is specified on the boundary surface S” [4,18,19,20]. Its mathematical form could be expressed as follows [4,18,19,20,21]:

where ρ _e is the charge density in the volume V outside the metal. Its mathematical proof is provided in many famous textbooks [4,18,19,20,21]. The first uniqueness theorem with Dirichlet boundary condition is valid in mathematics. According to the theorem, if the surface potential φ(r)∣s can be specified accurately, the solution φ(r) in the volume V can also be specified. However, for the application of first uniqueness theorem in electrostatic problems of the real metals, what is commonly known is the external electrostatic potential φ _e(r) which cannot solely give the electrostatic potential of the metal. The electrostatic potential may be dominated by both the external electrostatic potential φ _e(r) and the intrinsic electrostatic potential φ _i(r), as is indicated by Eq. (5). So, the corresponding surface potential of the metal should be given by

where φ(r)∣ _S stands for the precise electrostatic potential at the metal surface, and φ _i(r)∣ _S and φ _e(r)∣ _S denote the intrinsic electrostatic potential and external electrostatic potential at the metal surface, respectively. As indicated previously, φ _e(r)∣ _S does not exhibit position dependence and is a constant at the metal surface, but φ _i(r)∣ _S is usually position dependent. The intrinsic electrostatic potential φ _i(r) may be very hard to obtain, because the correlated Fermi level usually depends on many complex unspecified factors, e.g., the residual stresses, surface reconstruction, composition distribution, and the number and distribution of various defects. For the intrinsic electrostatic potential, it can be concluded that it may rest with the distribution of intrinsic electrostatic charges in the metal and it may satisfy the relation ∇ 2 φ i ( r → ) = 0 in the concerned volume V outside the metal. In another respect, when applying the first uniqueness theorem for a metal, one generally only considers the external electrostatic potential φ _e(r)∣ _s but neglects the intrinsic electrostatic potential φ _i(r)∣ _s at the metal surface. As a result, the uniqueness theorem in the actual electrostatic problems of a metal becomes

The solution is unique and the proof is simple and shown as follows. The functions φ _e1(r) and φ _e2(r) are supposed to be the solutions. Let the function U be U = φ _e1(r) – φ _e2(r). The function U satisfies the following dominant equation and Dirichlet boundary condition according to Eq. (15):

Using the identity equation for arbitrary scalar field ψ,

The function U satisfies the following equation:

Using the dominant equation and Dirichlet boundary condition for U, the above equation changes to

which implies ∇ U = 0 . Consequently, the function U is a constant inside the concerned volume V. For Dirichlet boundary condition of the function U, i.e., U = 0 at the metal surface, U must be zero inside the volume V so that φ _e1(r) = φ _e2(r) and the solution of Eq. (15) is unique. As is shown, the precise solution is the external electrostatic potential including the contribution of induction charges at the metal surface. But it does not contain the contribution of the intrinsic electrostatic potential φ _i(r) in the volume V, so it is only a rough approximation for the electrostatic problem. Under the approximation, the far-field solution may be relatively accurate because the intrinsic electrostatic potential φ _i(r) usually decays rapidly and almost vanishes in the far-field regions. However, the near-field solution may show a very noticeable deviation because the intrinsic electrostatic potential φ _i(r) may display a notable magnitude in the near-field regions. Therefore, the intrinsic electrostatic potential φ _i(r) and the related intrinsic electrostatic field E _i(r) must be taken into account when one will carry out the investigations on near-field physics and chemistry, especially the surface physics and surface chemistry. Here the words “far-field” and “near-field” refer to the concerned regions far away from the metals and the concerned regions in the vicinity of the metals, respectively.

2.1.4 Second uniqueness theorem for metals

The second uniqueness theorem states that “In a volume V surrounded by metals and containing a specified charge density ρ _e, the electric field is uniquely determined if the total charge on each conductor is given” [4,19,21]. Its mathematical form could be given by [4,19]

where Q denotes the total charge on a metal. The mathematical proof is available in many classical textbooks on electrodynamics [4,19,21]. As shown in Eq. (5), the solution for the total electrostatic potential φ(r) is composed of the external electrostatic potential φ _e(r) and the intrinsic electrostatic potential φ _i(r). Indicated by the MEEC, the metal may not be an equipotential body for the total potential φ(r), so the common mathematical proof [4,19,21] based on the EEC that the metal is an equipotential body may not hold right. The proof may be simple. The function φ ₁(r) is supposed to be a solution of Eq. (17). The function φ ₁(r) + aφ _i(r) (a is an arbitrary nonzero constant) must be another solution if the function φ _i(r) satisfies

One may also easily find that there are many nontrivial solutions for the problem described by Eq. (18), which is shown in Figure 3. In other words, the mathematical solution of the problem stated by the traditional second uniqueness theorem on metals is not unique and thus the second uniqueness theorem is not established rigorously in the viewpoint of mathematics.

Figure 3

The distribution of intrinsic charges at the totally neutral metal surfaces. (a) Little intrinsic charges at the metal surfaces with the corresponding electrostatic potential φ _i1(r); and (b) more intrinsic charges at the metal surfaces with the electrostatic potential φ _i2(r).

Despite the faults of the traditional second uniqueness theorem in mathematics, it could still be useful in physics. When the position-dependent intrinsic electrostatic potential φ _i(r) is unique and precisely known, the corresponding solutions φ _e(r) and φ(r) could be determined uniquely. In many real situations, a metal is intrinsically neutral and thereby the totally intrinsic electrostatic charge in the metal is zero, which means that the intrinsic electrostatic potential φ _i(r) satisfies Eq. (18). Simultaneously, as analyzed in Section 2.1.3, the intrinsic electrostatic charges may reside in the metal and thus the potential φ _i(r) may conform to the relation ∇ 2 φ i ( r → ) = 0 in the volume V outside the metal. As a result, the mathematical form of the second uniqueness theorem on metals changes into the following form by means of Eqs. (5), (14), and (18):

In view of the MEEC that the external electrostatic potential φ _e(r) is equipotential in the whole metal body, it could be found that the solution is unique based on the same mathematical proof method in the textbooks [4,19,21]. The related proof is offered here. The external electrostatic potential functions φ _e1(r) and φ _e2(r) are supposed to be the solutions. Let the function U be U = φ _e1(r) – φ _e2(r) and the function U satisfies the dominant equations

Using the identity equation shown by Eq. (16) the function U fulfills the following equation:

As pointed out by the MEEC, the external electrostatic potentials φ _e1(r) and φ _e2(r) are constant potentials at the metal surface, so the function U is also a constant at the metal surface S. As a result, the above equation can be written as follows:

Using the boundary condition of the function U, the above equation becomes

which implies ∇ U = 0 and U is a constant in the concerned volume V, meaning that the solution is unique apart from an unimportant arbitrary additive constant.

Therefore, the second uniqueness theorem for metals exactly describes the external electrostatic potential φ _e(r) instead of the total electrostatic potential φ(r). To be noted is that the discussed external electrostatic potential φ _e(r) contains the contributions of both the induction charges in the metal and the stationary charges in the concerned volume V outside the metal.

For the applications of the second uniqueness theorem in the actual electrostatic problems of metals, like the case of the first uniqueness theorem, people usually neglect the important role of the intrinsic electrostatic potential φ _i(r). Hence, the obtained solution may be the external electrostatic potential φ _e(r). It may be a relatively good solution in the far-field regions but need a careful correction in the near-field regions, as is in analogy with the case of first uniqueness theorem.

If the electrostatic problem is given by Robin’s boundary condition shown below

where α, β are two arbitrary real constants with the same sign. The solution is also unique in mathematics. The proof is simple and presented as follows. Let the functions φ ₁ and φ ₂ be two solutions. Let the function U be U = φ ₁ – φ ₂. The function U satisfies

Using the identity Eq. (16) the function U fulfills

Inserting Robin’s boundary condition will yield

Since α, β are two arbitrary constants with the same sign, the above equation holds right only when the difference function U is zero. Therefore, the solution is unique. When the electrostatic problems of metals with Robin’s condition is solved, the contribution of the intrinsic electrostatic potential φ _i(r) should be included.

In another respect, sometimes what is encountered is the electrostatic problem that exhibits the mixed boundary condition, i.e., Dirichlet boundary condition over a part of the metal surface S1 and Neumann boundary condition over the remaining part S2, as is represented by the following equation:

Its mathematical solution is also unique. The proof is almost the same as the proof for uniqueness theorem. To be noted is that the solution φ(r) is not an equipotential at the part of metal surface S1 and it should contain the contribution of intrinsic electrostatic potential φ _i(r). On the other hand, what the boundary condition gives is not the surface charge density at the remaining part of metal surface S2. If the electrostatic problem of metals is described by the mixed boundary conditions in the following mathematical form:

the solution is not unique in mathematics because the accurate potential φ(r) = φ _e(r) + φ _i(r) at the metal surface S2 is not a constant. And there are many solutions for the problem. The proof is similar to that of the traditional second uniqueness theorem for metals. On the contrary, if the problem is given by the mixed boundary condition shown below, then

the solution is unique because the external electrostatic potential φ _e(r) is a constant potential over the metal surfaces S1 and S2. In other words, what the traditional Unique theorem with the above mixed boundary condition uniquely gives is the external electrostatic potential φ _e(r) not the accurate potential φ(r).

The above discussions on uniqueness theorem (first uniqueness theorem and second uniqueness theorem) may not only point out the faults but also generate some new knowledge in related areas. On one hand, it may inspire people to realize the crucial role of the intrinsic electrostatic potential and the intrinsic electrostatic field in some scientific areas, particularly the surface physics and surface chemistry. On the other hand, the new understandings of the uniqueness theorem will subsequently give birth to new knowledge of the classical method of image charges and the famous electrostatic shielding in electrodynamics, which is addressed in the later part of the article.

2.1.5 Method of image charges

The method of image charges is popularly utilized to solve many electrostatic problems of metals. Its validity builds upon the uniqueness theorem. Its introduction and applications in electrostatic problems of metals could be found in some textbooks on electrodynamics [1,4,18,19,22,23]. As is demonstrated, if the solution not only satisfies Poisson’s equation in the concerned volume but also gives the correct EBC, it must be right. However, for the applications of the method in the related electrostatic problem, the commonly obtained solution may either satisfy Eq. (15) or Eq. (19), completely neglecting the contribution of the intrinsic electrostatic potential of the metal in the concerned volume. Therefore, as discussed for uniqueness theorem in the preceding sections, the solution that is obtained based on the method of image charges is the external electrostatic potential φ _e(r) rather than the precise electrostatic potential φ(r) shown in Eq. (5). Hence, the method of image charges may be an approximation. The approximation may be satisfactory in the far-field regions but it may need to be corrected in the near-field regions, which should be paid attention to for the surface-science researchers.

2.1.6 Electrostatic shielding

Electrostatic shielding can create a volume for protecting any concerned electric devices and instruments from the effect of any external electrostatic field. It is well-known and has been widely applied in various fields for more than 200 years up to now. For example, it was employed to accomplish the precise test of Coulomb’s law by many scientists in different generations, e.g., Cavendish (1773) [24], Maxwell (1873) [24], Plimpton and Lawton [25], and Williams et al. [26]. Wherein Williams et al. [26] achieved a miraculously precise upper limit on the deviation δ = (2.7 ± 3.1) × 10⁻¹⁶ which was commonly used to certify Coulomb’s inverse square law in the form 1/r ^2+δ .

In electrodynamics, the electrostatic shielding states that the electrostatic field is precisely zero in the empty cavity of a closed metal surface [1,21,27]. The conclusion builds on the first uniqueness theorem and the traditional EEC that the electrostatic field inside a metal must be null [27]. On the contrary, as concluded by the previous sections, the MEEC and uniqueness theorem for metals suggest that what must be null and unique is the electrostatic field induced by the external electrostatic potential, but the macroscopic intrinsic electrostatic field stemming from position-dependent Fermi level may be allowed to sustain inside the metal. As a consequence, the non-null intrinsic electrostatic field E _i(r) and related intrinsic electrostatic potential φ _i(r) may exert on the empty cavity of a closed metal surface, which is schematically shown in Figure 1. In other words, a real metal popularly presents net charges in some regions and displays a net electrostatic field in the empty cavity enclosed by the metal, but both of them are very difficult to eliminate. So, the electrostatic shielding can shield any external electrostatic field but cannot shield the intrinsic electrostatic field of the metal no matter how strong or weak the intrinsic electrostatic field is.

The question is why the intrinsic electrostatic field E _i(r) and intrinsic electrostatic potential φ _i(r) does not usually influence the related electrostatic shielding experiments. It may be attributed to the following reasons. First, the position-dependent Fermi level in a metal may not be altered under the commonly external electrostatic field and the related potential. So, the corresponding intrinsic electrostatic field E _i(r) would not vary in the metal and cannot give rise to the electromagnetic effect. Second, the concerned electric devices and instruments in the cavity may usually be put in the far-field regions where the intrinsic electrostatic field E _i(r) may be too weak to be detected. To the contrary, if the concerned devices would vibrate in the near-field regions of the cavity, an electrical induction current might emerge and a weak electromagnetic wave might be radiated simultaneously.

2.1.7 Thomson’s theorem

Thomson’s theorem states that “Static charge on a system of perfect conductors distribute itself so that the electric stored energy is minimum” [20], where the “perfect conductor” refers to “a material that has many free charges to move under external influences, electric or non-electric” [20]. The mathematical proof of Thomson’s theorem could be found in some textbooks on electrodynamics [18,20]. Thomson’s theorem and the corresponding proof only considers the external electrostatic potential φ _e(r) but omits the vital role of the intrinsic electrostatic potential φ _i(r) and the intrinsic electrostatic field E _i(r). Considering both the electrostatic potentials, the total electrostatic energy stored in the system should be given by

where W _t denotes the totally stored electrostatic energy in the metal systems, V _m is the volume of the metal systems, and V is the concerned volume outside the metal systems. It could be written as the summation of the following three types of electrostatic energies:

where W _i stands for the intrinsic electrostatic energy contributed by the intrinsic electrostatic potential φ _i(r) and the intrinsic electrostatic field E _i(r), W _e is the electrostatic energy arising from the external electrostatic potential φ _e(r) and the external electrostatic field E _e(r), W _ie is the interaction energy between the two fields.

According to the MEEC in Section 2.1.1, the potential φ _e(r) is constant and the field E _e(r) is null inside the metal, i.e., ∇φ _e(r) = 0. So, the interaction energy W _ie could be expressed as follows:

The intrinsic electrostatic charges are distributed within the metal such that the potential fulfills the relation ∇² φ _i(r) = 0 in the volume V outside the metal systems. Using the equation ∇φ _e(r) · ∇φ _i(r) = ∇[φ _e(r) ∇φ _i(r)] − φ _e(r) ∇² φ _i(r), the interaction energy W _ie may be written as follows:

where Q _i is the totally intrinsic charges in the metal, and S is the closed surface of the volume V outside the metal systems. The non-charged metal may inherently be neutral and the totally intrinsic charges inside the metal may be zero, i.e., Q _i = 0. Thus, the interaction energy W _ie may be zero. In general, during the charging or discharging processes of a metal, the yielded electric force exerted on the metal may be so weak that the microstructure and Fermi level may not vary. Therefore, the intrinsic electrostatic energy W _i may be an invariable quantity for the definite metal systems. As a consequence, the distribution of charged charges at the metal surface influences the incremental electrostatic energy W _e.

It may be easily found that if the metal is an equipotential body for the external electrostatic potential φ _e(r), the stored electric energy W _e is minimum. The corresponding proof may be the same as that in many textbooks on electrodynamics [18,20].

According to the above discussions, the famous Thomson’s theorem may be modified as “Charged net charges on a system of real metals distribute themselves so that the incremental electric energy is minimum.” This modification may bring several merits. First, unlike Thomson’s theorem applicable for perfect conductors, the modified Thomson’s theorem may be valid for both perfect conductors and real metals. In other words, the application scope of the modified Thomson’s theorem is much larger than that of Thomson’s theorem. Second, considering the existence of intrinsic electrostatic field and the stored electric energy, the statement of the modified Thomson’s theorem may be more accurate than that of Thomson’s theorem.

2.1.8 Green’s reciprocation theorem

Green’s reciprocation theorem is usually expressed in the following form [18,20]:

where φ _k is the electrostatic potential of the kth metal when it carries charge q _k , and φ k ′ is the potential when it carries charge q k ′ . The proof could be available in the textbooks on electrodynamics [18,20]. Green’s reciprocation theorem is very useful in solving many electrostatic problems of metals. However, as revealed by MEEC, the position-dependent intrinsic electrostatic potential φ _i(r) was usually omitted for a metal and the metal may not be an equipotential body for the total electrostatic potential. To one’s surprise, it means that any constant potential cannot accurately describe the total electrostatic potential of a metal. So, the popular form of Green’s reciprocation theorem may not be appropriate and has to be modified for wide applications. In actual applications, the known potential of a metal is usually the external electrostatic potential φ _e(r). It is a constant potential for the metal and satisfies the relation ∇² φ _e(r) = 0 outside the metals, so the Green’s reciprocation theorem may better be modified as

where φ _ek is the external electrostatic potential of the kth metal when it carries charge Q _k , and φ e k ′ is the external electrostatic potential when it carries charge Q k ′ . The mathematical proof may be the same as that in the related textbooks [18,20].

The modified Green’s reciprocation theorem could be utilized to solve the related electrostatic problems of metals. By modifying Green’s reciprocation theorem, we can obtain the external electrostatic potential rather than the total electrostatic potential of the metal body. If the investigated topics are associated with physics and chemistry at metal surfaces, the intrinsic electrostatic potential of the metal must be taken into account seriously.

2.2.1 Electric transport properties at surfaces

As shown in Figure 4, when an external voltage is applied to the metal surface with different Fermi levels, an electrical circuit may be formed. In analogy with the situation for the metal with different Fermi levels [5], the electrical current–voltage (I–V) relation of the metal surface may exhibit rectifying characters. If a forward bias external voltage is applied as shown in Figure 4(a), the energy of free electrons in region III may be uplifted relative to that of free electrons in region I, so that free electrons would drift from region III to region I, creating an electrical current which is displayed in Figure 4(c). However, upon application of a small backward bias voltage shown in Figure 4(b), the electrical current may be almost zero except a tiny tunneling current, because the free electron movement may be blocked by the energy barrier originating from the Fermi level difference. If the backward bias voltage increases to a critical value V _Z , the free electrons in region I may gain enough energy to surpass the energy barrier and reach region III easily, thereby forming a notable electrical current, as shown in Figure 4(c). And the critical voltage V _Z could be obtained according to Eq. (1) as follows:

where e is the electron charge, and E F ( r → III ) and E F ( r → I ) are the Fermi levels of region III and region I, respectively. It means that in the presence of TEF, the metal surfaces sometimes could rectify electrical current and behave as a p–n junction. And the rectifying characters may be more obvious at low temperatures because of the suppression of thermal excitation.

Figure 4

Electrical current–voltage (I–V) relation for the metal surface with a TEF. Region I (colored yellow) and region III (colored magenta) denote places with lower Fermi level and higher Fermi level, respectively, while region II shows the transition zone between region I and region III. (a) forward bias voltage; (b) backward bias voltage; and (c) I–V characteristics.

For actual metal surfaces, the complex strains, adsorptions, defects, and so on may induce very complex TEF. As revealed, the fields could cause the rectifying characters and usually reduce the surface electrical conductivity. It may enlighten people that to obtain a relatively precise electrical conductivity of metal surface, one had better employ a large electrical voltage to measure in both ways of forward bias and backward bias and take the larger value of conductivity.

2.2.2 Rashba effect at the metal surface

The Rashba effect first discovered in 1959 describes a momentum-dependent spin splitting of electron states in a bulk crystal [28] and low dimensional electron systems [29]. For an electron system with space inversion-symmetry breaking, it is commonly described by the following Hamiltonian:

where k → is the electron momentum, σ ˆ is the Pauli matrices, n → is the unit vector normal to the surface, and the parameter α _R is Rashba coupling which is usually proportional to the magnitude of electric field.

Based on the physics of Rashba effect, the TEF at metal surface may also contribute to the momentum-dependent spin splitting of electron bands, and the correlated Hamiltonian may be given by

where η is the coupling parameter with the unit m². Appearance of this Hamiltonian may further reduce the symmetry of electron spin bands in the momentum space. To examine its effect, the total Hamiltonian of the electrons should contain the kinetic energy of the electron, the conventional Hamiltonian for Rashba effect, and the Hamiltonian term H _RZ originating from the TEF at the metal surface. So, the total Hamiltonian could be written as follows:

where the first term on the right hand is the kinetic energy of the electron and p signifies the momentum operator. In terms of simple calculation, the eigenenergy is as follows:

It shows that the momentum-dependent splitting of electron band may be shifted in the presence of TEF. When the tangential field would vanish, the related eigenenergy would reduce to the familiar solutions for the Rashba Hamiltonian, as inversely verifies the correctness of the above electron bands given by Eq. (28).

Due to the TEF in Eq. (26), a spin component normal to the surface will appear as

where |φ ₁ 〉, 〈φ ₁| is the eigenket and eigenbra, respectively, and S _Z is the spin component normal to the surface.

Generally speaking, the magnitude of tangential field in many metals may not be so strong that its effect might be weaker than Rashba effect. But for some special situations where the Rashba Hamiltonian vanishes, i.e., the zero Rashba coupling α _R, the eigenenergy for the 2D electron systems would be

The effective magnetic field is

where g is the spin-g factor and has a value of about 2, and μ _B is the Bohr magneton. And the related electron spin component along surface normal is

where | φ 1 ′ 〉, 〈 φ 1 ′ | is the corresponding eigenket and eigenbra, respectively. Very interestingly, for a 2D electron system even without magnetic field, if the electrons only possess the momentum parallel to the surface and the normal component is zero, the presence of any TEF would give birth to a totally spin-polarized state and the electron spin polarization strongly depends on its momentum. At ground state, the electron spins are polarized and parallel to the effective magnetic field B _eff as shown in Figure 5. The effective magnetic field and related spin splitting energy might be detected by means of electron spin resonance experiments which were successfully carried out to measure the spin splitting energy for a 2D electron system [30]. For such a 2D electron system as displayed in Figure 5(a), the up spins would accumulate on the left side, while the down spins would get together on the right side of the metal surface. On application of an external electrical voltage at metal surface, only the electrons with up spin are allowed to flow but the electrons with down spin are inhibited as illustrated by Figure 5(b). Hence, the metal surface with the TEF might be a platform for anomalous quantum Hall effect.

Figure 5

Momentum dependence (short black arrows) of polarized electron spins under a TEF at the metal surface. The tangential field is along the x axis as indicated by the long black arrows. The red circles present the electrons. Red arrows and blue arrows denote the polarization direction of electron spins. (a) without external electrical voltage; and (b) with an electrical voltage V _e.

2.2.3 Effect on the photoemission spectroscopy (PES)

According to electrostatics [1,4], i.e., circumstances in which no physical variables depend on time, one may have

So the TEF along the metal surface may cause a related tangential component outside the surface as illustrated by black arrows in Figure 2. It may have an important effect on PES.

Based on the physical theory of PES [31], a tangential field may affect PES in two ways. One is that it may change the detected kinetic energy of photoelectrons by the quantity.

where ΔE _kin is the variation in kinetic energy of photoelectron, e is the electron charge, and V ₀ is the potential difference between the positions of photoelectron emission point and electron analyzer. As shown in Figure 6, because of the complexity for both the experimental design and the unknown distribution of TEF in real experimental metal surface, the precise value of kinetic energy variation may be difficult to get. But one may estimate its value as follows:

Figure 6

Sketch of PES affected by TEF at the metal surface. When a light beam (colored purple) radiates at the metal surface, some photoelectrons (red dots) would be emitted and their motion would be influenced by the electrostatic field near the surface.

The other is that for the photoelectrons emitting out of the surface, the tangential field may result in alterations for their wave-vector component parallel to the surface. It is

where m _e is the electron mass, ħ is the reduced Planck’s constant, k || i and k || f , respectively, denote the wave-vector component parallel to the surface for the initial state and final state of a photoelectron, and V _|| is the potential difference between photoelectron emission point and electron analyzer and it is V || = − ∫ r f r i E → || ( r → ) ∙ d r → , where E → || ( r → ) is position-dependent TEF outside metal surface, and r _f, r _i present positions of electron analyzer and photoelectron emission site, respectively. As indicated, the tangential field can indeed change the wave-vector component parallel to the surface. Despite lack of knowledge on the detailed electrostatic field near the metal surface, an estimation could be made and the alteration may be in the range

In view of Eqs. (35) and (37), one may find that a larger Fermi level difference and thus a stronger TEF at the metal surface usually have a more notable effect on the PES. Even for the single-crystalline surface at a uniform temperature, there still exists the unavoidable strains arising from residual stresses and defects. For instance, the residual stresses are often reported to reach ∼10² MPa [13,14,32,33], which can yield a few thousandths of a strain [34]. For most metals, the mechanical-electric coupling strength |∂E _F/∂ξ _ii | (ξ _ii is the main strain) may be in the range 1–10 eV. So the residual strains at the metal surfaces could cause the order of 10 mV for the surface potential variations, which is consistent with the experimental observations [8,35]. The magnitude of the surface potential variations may place a noticeable restriction on the measurement precision of PES, which deserves attention in the related experiments, e.g., angle-resolved photoemission spectroscopy (ARPES).

The existence of the TEF at the metal surface may hybridize the electron orbital wave functions, for instance, hybridization of s and p orbitals. According to the quantum perturbation theory, the hybridized s and p wave functions are (to first-order precision) as follows:

where Ψ _ns and Ψ _np are the electron wave functions with main quantum number n and angular momentum s, p, and Ψ ns ′ , Ψ np ′ stand for the corresponding hybridized wave functions, respectively.

This hybridization may affect the PES of the metal surface. If the TEF is along x axis, the hybridization only exists between s and p _x electron orbitals. So, the corrected p _x wave function has a component of s orbital wave function. For a tetragonal metal surface with conductive p electrons, a beam of photons can be utilized to radiate the metal surface at a very small glancing angle (e.g., less than 5°) as shown in Figure 7. According to the ARPES theory [36], the measured ARPES intensity may be proportional to the square | ⟨ Ψ f | A → ⋅ P → | Ψ np ⟩ | 2 , i.e., I ∝ | ⟨ Ψ f | A → ⋅ P → | Ψ np ⟩ | 2 , where A is the electromagnetic gauge and P is the electron momentum. Considering the photon radiation at the metal surface with tetragonal symmetry, the symmetry analysis could be used to clarify the contributions to the APRES intensity. If the glancing photons have s polarization (the photon electric field perpendicular to the scattering plane k _x –k _z ), the ARPES intensity contributed by the scattering matrix element is zero along k _y direction | ⟨ Ψ f | A → ⋅ P → | Ψ n p x ⟩ | 2 = 0 , because both the final state of photoelectron wave function Ψ _f and the term A → ⋅ P → lying in the mirror plane k _y –k _z are even, but Ψ _npx is odd relative to the mirror plane k _y –k _z . However, the presence of TEF along x axis may generate the hybridization between s electron orbital and p _x electron orbital, which enables the term | ⟨ Ψ f | A → ⋅ P → | Ψ n p x ′ ⟩ | 2 > 0 to contribute to the ARPES intensity along k _y axis. In the related experiments, one may find that when the incident photons having s polarization are scattered in the k _x –k _z plane, the corresponding ARPES intensity along k _y axis may be more or less stronger than that along k _x axis.

Figure 7

Schematic diagram of a beam of photons (denoted by red line) with s polarization (photon electric field perpendicular to the scattering plane k _x –k _z ) or p polarization (photon electric field lying in the scattering plane k _x –k _z ) radiating at a metal surface with tetragonal symmetry. The TEF indicated by blue arrows is along k _x direction.

2.2.4 Effect on the scanning tunneling microscope or spectroscopy (STM/STS)

The altered Fermi level and the correlated TEF may have a non-negligible influence on the STM/STS. And the tunneling current may be changed by the induced variations in the density of state (DOS) and work function. For a region with altered Fermi level at a metal surface, a tunneling current may appear when an electrical bias voltage V _bias would be applied at low temperatures. The current could be written as follows [37,38]:

where M is the tunneling matrix element, N _t(E _F0) is the DOS at the Fermi level E _F0 of the tip, N _s(ε) is the energy ε dependent DOS of the sample surface confronting the tip, and f is the Fermi-Dirac distribution function. The local DOS of the sample may be a slowly varying function, so Eq. (40) could be simplified to

where E _F is the altered Fermi level of the region, and V _Z is the electric potential difference between this region and surrounding regions with initial Fermi level E _F0 as shown in Figure 8, and it is E _F + eV _Z = E _F0. Based on Eq. (41), one may find that for the region with altered Fermi level, the tunneling current may be determined by its local DOS at the corresponding energy level (E _F + eV _Z) instead of its local DOS at the Fermi level E _F.

Figure 8

A bias voltage V _bias between tip of STM and metal surface with the up-shifted Fermi level colored by magenta. The up-shifted Fermi level may make this region possess a positive electric potential V _Z and the surrounding electric field is shown by black arrows.

On the other hand, owing to the alteration in Fermi level, the work function of the electron in this region may be changed

where ϕ _s0 denotes the work function of the metal surface with pristine Fermi level, and ϕ _s is the work function of the metal surface with altered Fermi level. The variations in the work function may further lift the tunneling current by changing the key parameter, i.e., inverse decay length, because the tunneling current is I ∝ e − 2 κ d , where κ is the inverse decay length and d is the distance between tip and sample surface [39]. The modified inverse decay length may be given by [40,41]

where W is the energy of the state relative to the Fermi level E _F. As is shown in this equation, the regions of metal surface with up-shifted Fermi level would be expected to yield an enhanced tunneling current and vice versa. These regions may be detected by the constant height mode of operation for STM.

2.2.5 Surface transport and segregation

Analogous to the case of metals [42], the TEF at the surface of metallic terminal solid solution (MTSS) may also cause a composition segregation. For simplicity, the two-component MTSS could be investigated as an example. For the minority atoms at MTSS surface, their diffusion flux could be given by the famous Fick’s law [43].

where n(r, t) is the position r and time t dependent minority atom concentration, and D is the diffusion constant at the surface. Based on Einstein diffusion relation, the diffusion constant equals to D = uk _B T, where k _B is the Boltzmann constant, T is the temperature, and u is the mobility of minority atom at the surface.

In view of the possible TEF at the surfaces, the field-induced another transport mechanism may play an important role in the evolution processes of the surface composition. Despite many factors that enable the Fermi level to shift and thereby induce a related electric field, only the electric field originating from the inhomogeneous strain is considered. By employing Yuheng Zhang equation, the electric field generated by the non-uniform strain at a surface normal to z axis may be

where ξ _zi (i = x, y, z) is the strain components at the surface normal to z axis and the subscript indices obey Einstein summation convention, and dE _F/dξ _zi denotes the mechanical-electric coupling of the MTSS. According to Teorell’s theory [44], the electric field would result in a correlated drift term and it is

where Q denotes the net charges carried by a minority atom due to the electronegativity differences between the minority atoms and the majority atoms.

In terms of Eq. (1), the inhomogeneous distribution of minority atoms at the MTSS surface may also give rise to a tangential electric field along the surface and it may be

where e is the electron charge, and E _F is the minority atom concentration dependence of Fermi level. Likewise, this field may create another drift flux of the minority atoms and it could be given by

The total transport flux of the minority atoms at the TMSS surface may be the summation of the diffusion term and the two drift terms.

where the parameter is α = Q(dE _F /dn)/ek _B T. When the equilibrium state is approached, the total transport flux of the atoms at the MTSS surface may be rigorously zero. So, the distribution of the minority atoms at the MTSS surface could be obtained analytically

where the mathematical function is defined as ProductLog(λe ^λ ) = λ, the Fermi level difference ΔE _F stemming from strains is ΔE _F = E _F(r) − E _F(r ₀). It can be seen that the existence of a TEF at MTSS surface may induce the surface composition segregation, which may further influence the physical properties and chemical properties of the surface. If the magnitude of the product is very small |αn(r)| ≪ 1, this equation could be simplified to

It may indicate that the composition distribution at the MTSS surface may be controlled by the temperature T, the electronegativity difference, and the shifted Fermi level caused by the strains. At high temperatures, the composition distribution at the surface may be much more uniform than that at low temperatures. A smaller electronegativity difference between the minority atoms and the majority atoms may lead to a relatively uniform composition distribution and vice versa. In general, the continuous solid solutions [45] listed in Table 1 may exhibit a much more uniform composition distribution, whereas the discontinuous solid solutions especially those with small solid solubility usually display serious composition segregation. This may be consistent with investigations on the surface segregation of bimetallic nanoparticles [46]. Another important factor is the Fermi level difference that usually originates from the popular strains at the MTSS surface because of the residual stresses and various types of defects such as dislocations. Here a preliminary estimation may be helpful and necessary. The residual stresses could usually reach several hundred MPa [13,14,32,33], so the residual strain may be several thousands. For most MTSS, the mechanical-electric coupling strength may be in the range of 1–10 eV, thereby creating a Fermi level difference ΔE _F in the range of 5–50 meV at the surfaces, which agrees with the experimentally observed magnitude of the surface potential variations [5,6,7]. If the net charge carried by a minority atom is an electron charge, the room-temperature-composition segregation ratio at the surface might reach 1.2–7 according to Eq. (51), which may be in qualitative accordance with the segregation ratio induced by the enrichment of Pt atoms at edges and corners of the nanoparticle surfaces [47,48]. In another respect, the more commonly observed is the radial strain reaching ∼1% for many nanoparticles, so a Fermi level difference ΔE _F of 10–100 meV may be yielded along the radius of the nanoparticles. If the net charge carried by a minority atom is also an electron charge, the room-temperature segregation ratio estimated by Eq. (51) may be in the range of 1.5–50, which is consistent with the experimental investigations on segregation ratio in bimetallic nanoparticles [46,47,48,49]. The above estimations may prove rationality of Eq. (51) in spite of their roughness. To get much more precise segregation ratio by using Eq. (51), one should obtain more accurate parameters Q and ΔE _F.

2.2.6 Effect on field emission

Field emission is an important technique and has been applied in numerous areas such as the field emission microscopy. Under an externally applied strong electric field shown in Figure 9(a), the electrons in a cold metal can tunnel into vacuum and form an electrical current. According to the famous field emission model proposed by Fowler and Nordheim [50], the current is

where ε _F is the Fermi energy, i.e., the largest kinetic energy for free electrons in a metal at zero temperature, ϕ is the work function, E is the applied intense electric field at the metal surface, and e presents the electron charge.

Figure 9

Sketch of field emission for a metal. (a) electron emission for the metal with a work function ϕ under an intense field E at the metal surface; and (b) electron emission for the region with a shifted Fermi level by the amount ΔE _F and the related electric field is changed to be E–E ₀ where E ₀ results from the shifted Fermi level.

The field emission process of a cold metal may be affected by the existence of TEF at the metal surface from two respects. As displayed by Figure 9(b), one is the shifted Fermi level whose variations may correspond to the changes in work function at the low temperatures. The other may be the weak modifications on the intense electric field at the metal surface. The updated electrical current due to field emission may be given by

where ε F ′ is the shifted Fermi energy, ΔE _F is the lifted amount of Fermi level, and E ₀ denotes the electric field arising from the shifted Fermi level and it may be much weaker than the applied intense field.

As shown by this equation, the up-lifted Fermi level region at the metal surface may generate a larger field emission current, because the related work function may be weakened. Whereas the down-lifted Fermi level region may cause a smaller field emission current. For the material surfaces such as the nanoparticle surfaces, the experiments and simulations indicated that a smaller radius of curvature can usually yield a larger local strain [51,52,53,54,55,56,57]. According to the mechanical-electric coupling in metals, i.e., Yuheng Zhang effect, the noticeable local strain may lead to an apparent shift in the Fermi level. Therefore, it could be inferred that both the intense field and up-shifted Fermi level due to notable local strain play an important role in the field emission of a sharp tip with a radius of curvature in nanoscale.

2.2.7 Micro-earth

The experimental and theoretical investigations indicated that a dramatic and widespread strain gradient exists along the radius of nanoparticles [51,52,53,54,55,56,57]. According to the Yuheng Zhang effect and Yuheng Zhang equation [9,10], this strain gradient in the nanoparticles may give rise to a corresponding electric field along their radius, and it could be given by ref. [11]

where ξ _ij is the strain components. For a nanoparticle with a radius of several nanometers, the internal radial electric field may approach a magnitude ∼10⁸ V/m. Hence, a noticeable charge separation may appear in the nanoparticle and a larger number of charges could accumulate at the nanoparticle surfaces. For simplicity, this electric field is assumed to be a constant and the charge density distribution within the metal nanoparticles may be approximated as that in the core of the earth [58].

where r ₀ is the radius of the metal nanoparticle, ρ(r) is the radius dependence of charge density, and the parameter D is the electric displacement D = ε ₀ εE _n and its magnitude may reach a value 10 C/m² for some metal nanoparticles. Analogous to the possible creation of the earth’s geomagnetic field [58], the fast rotation of such a metal nanoparticle from west to east (the same rotational direction as the earth) will generate a correlated magnetic field which may be described in cylindrical coordinate system as shown in Figure 10.

where R is the distance from center of the metal nanoparticle, T is the period of rotation, μ ₀ is the magnetic susceptibility outside the nanoparticle, ψ is the polar angle from positive z-axis, and φ is the azimuthal angle in the x–y plane from the x-axis.

Figure 10

Schematic diagram of a magnetic dipole field yielded by fast rotation of a metal nanoparticle.

These equations display that the yielded magnetic field is a magnetic dipole field. For a metal nanoparticle with an average radius of 2 nm, if the period of its fast rotation could be 1 ps, a magnetic dipole field may be created around the nanoparticle and the field strength at the equator may reach ∼200 Gauss. The electric field along the radius of the metal nanoparticles may be assumed to be a constant, but the actual electric displacement may not be a constant and varies with the position. So, the above calculations could mainly predict the mechanism of magnetic field yielded by the fast rotation of a metal nanoparticle and a more precise determination of the magnetic field would still depend on a more precise position dependence of electric displacement vector.

In spite of the remarkable size difference between the metal nanoparticle and the earth, the fast rotation of a metal nanoparticle may be very like the earth considering the following points. First, a notable charge separation exists along the radius for both the metal nanoparticle and the metallic core of the earth due to Yuheng Zhang effect. Second, according to the possible mechanism of the earth’s geomagnetic field [58], the fast rotation of both the metal nanoparticle and the earth could generate a magnetic dipole field. Taking these similarities into account, the rotating metal nanoparticle may be named as “micro-earth.” How can people give birth to such a fast rotated micro-earth? A possible method might employ the incidence of photons with an angular momentum. The effective transfer of the angular momentum of photons to a nanoparticle may enable it to rotate rapidly and yield a micro-earth. In the future, the micro-earth may be utilized in some important areas such as micro-detection.

2.2.8 Adsorption of molecules at nanoparticle surface

As indicated in Section 2.2.7, the popular strain gradient in the nanoparticles usually generate a noticeable charge separation and the charges may agglomerate at surfaces especially the sharp edges and the shared corners as shown in Figure 11. As a result, an intense electrostatic field may be created in the vicinity of the nanoparticle surface. The field may sensitively depend on the position and decay rapidly with the distance away from the surface. In general, the spatial distribution of the electrostatic field may exhibit the following properties:

where r is the distance away from the nanoparticle surface, l ₀ denotes the surface strain relaxation length and is usually the length of several atomic layers ∼1 nm [53,57,59], and R ₀ is the radius of the nanoparticle. The electrostatic field may approach a magnitude ∼10⁸ V/m for the molecules adsorbed on the surface and even reach ∼10⁹ V/m near the sharp edges and shared corners. The existence of an intense electrostatic field affecting the thermodynamic behaviors of gas adsorption heavily can be expected. For the molecule system, the Gibbs energy change is as follows:

where G is the Gibbs energy of the molecule system, n = N/V is the concentration of molecules, V is the volume per mole molecules, N is the Avogadro constant, p is the pressure of the gas molecules, E is the electric field, p _e denotes the intrinsic dipole moment of an individual molecule, and 〈p _e〉 is the thermal average value and it could be given by

where E(r) is the position-dependent electric field, the parameter k _B is the Boltzmann constant, T is the temperature, and α _e stands for the electric polarizability of a molecule. Generally speaking, the polarizability α _e sometimes exhibits anisotropy and thus may be described better by a matrix. But it could be assumed to be a constant for simplicity in the following discussions.

Figure 11

Sketch of a cutaway drawing for a nanoparticle with charge separation. The strain (magenta zone and the strain relaxation length l ₀) due to surface relaxation may exist along the radius and could cause the related remarkable charge separation. The net charges at the surface may mainly be distributed at the sharp edges and shared corners.

When the neutral molecule system reaches thermal equilibrium, the spatial gradient of Gibbs energy must be zero and the equilibrium equation is as follows:

In the above calculations, the field effect on the electric dipole–dipole interaction is ignored for simplicity. Using Van der Waals equation [60,61] for 1 mole gas molecules,

where a is a parameter arising from the attractive interaction between molecules [60,61], b is the parameter due to the finite volume of a molecule, and R is the gas constant.

Substitute Van der Waals equation in Eq. (60),

Its analytical solution may be

where n ₀ is the concentration of molecules far away from the nanoparticles. In most cases, the relation nb/N ≪ 1 holds, meaning that the volume of molecules in gas state may be far larger than that for the same number of molecules in liquid state. Thus, the equation may be approximated as

Let the important parameter λ be

The above equation could be written as follows:

Its precise solution could be given by

where the function is ProductLog(x·e ^x ) = x. It indicates that the concentration of the adsorbed molecules at the nanoparticle surface may rely on several important factors such as the temperature, the magnitude of electric field, and the electric dipole of a molecule. The temperature has an important effect and the concentration increases as the temperatures drops. A more intense electric field at the nanoparticle surface would lead to a larger concentration. In addition, a bigger electric dipole of the molecule may also result in a larger concentration and vice versa. When the concentration of the adsorbed molecules at the nanoparticle surface reaches the saturation vapor concentration, the liquefaction may happen.

Considering the possible liquefaction in the vicinity of the nanoparticle surface, the whole adsorption around the nanoparticles may be contributed by two typical regions. The first is the region where the electric field is weak and decays rapidly with distance from the nanoparticle surface, thereby corresponding to the region full of molecules in the gaseous state.

In most situations, the relation may be valid |λn _s| ≪ 1 where n _s is the saturation vapor concentration, so Eq. (66) could be simplified to

The concentration of the adsorbed gas molecules near nanoparticle surface may be approximately proportional to the concentration n ₀ far away from the nanoparticles. The approximate relation between the gas pressure and the concentration of the gaseous molecules could be taken as n 0 ≈ p 0 / k B T , where p ₀ is the gas pressure. In reality, this approximation is the equation of state for the ideal gas. Thus, the adsorbed mass of the gaseous molecules could be given by

where m is the mass of an individual molecule, M _a is the total mass of the adsorbed molecules. It shows that the mass of the adsorbed molecule may be proportional to the gas pressure and the proportionality constant may become large at low temperatures, which is usually observed in experiments when the gas pressure is very low [62]. At relatively high temperatures, the relations p e E ( r → ) / k B T ≪ 1 and α e E 2 ( r → ) / k B T ≪ 1 may be fulfilled so that Eq. (59) could be approximated to be 〈 p e 〉 E ( r → ) / kT ≈ p e E ( r → ) kT 2 3 + α e E 2 ( r → ) / 2 k B T . As a result, the adsorbed mass given by Eq. (68) could be written as follows:

As is demonstrated, the adsorbed mass of the gaseous molecules may be proportional to the electrostatic energy around the nanoparticles. And the adsorbed quantity would be larger for the molecules with a bigger dipole moment p _e and a bigger polarizability α _e.

In another case λ n s ∼ 1 , the relation between the concentration n under the electric field and the concentration n ₀ may exhibit a concave hyperbolic curve given in Figure 11. This case may be rarely encountered in experiments because it requires the condition λn _s ∼1 that most kinds of gas molecules cannot satisfy.

Interestingly, if the parameter λ displays a negative value and the relation − λ n s ∼ 1 exists, the concentration n ₀ versus the concentration n may present a convex hyperbolic curve as shown in Figure 12. Considering the required negative value of the parameter λ, it may also be difficult for this case to be observed in experiments.

Figure 12

The reduced concentration n/n _s (n _s is the saturation vapor concentration) of adsorbed gas molecules in the vicinity of nanoparticle surface vs the reduced concentration n ₀/n _s far away from the nanoparticles. The black squares denote the curve with parameter λn _s = 0.05, the red diamonds denote the curve with parameter λn _s = 0.5, and the blue circles denote the curve with parameter λn _s = −0.5.

The second region is the liquefaction region where the molecule concentration reaches the value of liquid state, i.e., n = n _l, and its volume could be denoted as V _l. At the boundaries between the liquefaction region and gaseous region, the concentration of the gas molecules may reach the saturation vapor pressure p _s which corresponds to saturation vapor concentration n _s,

Combining Eqs. (59) and (70), a corresponding saturated electric field (SEF) may be uncovered.