Abnormal driving forces induced by curvatures and gradients of curvatures on micro/nano curved surfaces

2.1 The most simplified biomembrane mechanics

When deriving Eqs. (1)–(3) about ten years ago, the author had no motivation to aim at Pauli’s devil but was just directed by a very pure idea, i.e., to express the basic equation of biomembrane mechanics in the simplest form possible. At that time the author noticed that the basic equation of biomembrane mechanics was extremely complicated. Even the equilibrium equation for the simplest case of an axial symmetric biomembrane would cover half a paper. Why was the equation so complex? Was there anything wrong in the equation? After carefully checking, the equation was proved to be correct. Then where did the complexity come from? After lengthy considerations, several possible reasons were identified, and the space form finally drew the author’s attention.

Biomembranes at micro/nanoscale are very highly curved soft matter spaces. Such matter spaces may be abstracted geometrically as curved surfaces. Historically, there were two different paths for mathematicians to study curved surfaces. One was presented by Euler. He regarded a 2D curved surface as a subspace inserted in a 3D Euclidean space. Through the measure of the flat Euclid space, he determined the invariant properties of the curved surface. Another approach was presented by Gauss. He treated the curved surface as an independent space form. Through the measure of the curved space, he described the invariant properties of the curved surface. In other words, Euler took the Euclidean space as a reference space, and then he stood in the reference space and studied the curved surface. Gauss abandoned the reference space completely. He went into the curved space directly to examine the curved surface properties.

It has been proven that Euler’s path is narrow and rough, and it quickly approaches a dead end. Gauss’s path is wide and smooth. After Riemann’s revolutionary extension, Gauss’s path led to the golden road of the Riemann geometry. Two different mathematical personalities accepted different space viewpoints, chose different space forms, and developed different geometry roads.

Mechanics is controlled by geometry [12]. Biomembrane mechanics is essentially the mechanics of curved surfaces. Two different paths adopted in studying the curved surface may lead to two different biomembrane mechanics. The biomembrane mechanics based on a Euclidean space viewpoint will inevitably lead to complexities. The question is then whether the biomembrane mechanics based on the Riemann space viewpoint will lead to simplicity. To one’s surprise, the answer is negative.

After systematic analysis, we find that the Riemann space form is only the necessary condition for the simplification of biomembrane mechanics. In other words, with the Riemann space form introduced, biomembrane mechanics is not becoming inevitably simplified. To establish the most simplified biomembrane mechanics, we should not only give up the Euclid space form and accept the Riemann space form but also abandon Gauss parameter coordinates and introduce non-coordinate expressions in differential equations.

How do we realize non-coordinate expressions? The answer is to introduce differential operators. Differential operators have two contradictory sides. On the one side, differential operators must be defined through coordinates, i.e., they include partial derivatives with respect to coordinates. On the other side, once differential operators are defined, they are irrelevant to coordinates at all. Such irrelevance is called “invariance with respect to coordinate transformations” in mathematics. “Invariance” or “invariability” also means “objectivity”, which is a fundamental principle in mechanics. In other words, a differential-operator-based differential equation is the only choice for biomembrane mechanics to eliminate the superficial effect of coordinates.

Based on the above considerations, the pair 〚∇, ∇¯〛 was defined, and the simplified form of the differential equation for biomembrane mechanics, i.e., Eq. (1), was derived.

2.2 Inevitable inquires

In Eq. (1), the differential operator pair 〚∇, ∇¯〛 is an essential quantity, which has been systematically discussed. It was established in [1] that there are two fundamental and independent differential operators existing in the physics and mechanics on curved surfaces. It was demonstrated in [2] that 〚∇, ∇¯〛 is the best pair of the fundamental differential operators. In [3, 4], it is shown that 〚∇, ∇¯〛 may lead to a symmetric substructure of differential geometry, and this symmetric substructure will completely control the micro/nanophysics and mechanics on highly curved surfaces. All these papers are focused on the following key point: What invariant properties may 〚∇, ∇¯〛 possess? But another key point was not addressed: What is the meaning of 〚∇, ∇¯〛 in physical systems? This question is equivalent to “what is ∇¯(⋯)”, because the answer for “what is ∇” is clear.

2.3 Geometric meaning

In the term ∇⋅(∇ϕ,H/2+∇¯ϕ,K) in Eq. (1), two subterms are summed up. In mechanics, such subterms must have identical dimensions and similar meanings. Therefore, if ∇(···) is the gradient operator, then ∇¯(…) can also be termed the gradient operator. Although they both are gradient operators, their meaning is different: ∇(···) has no relation to curvatures and existed in classical geometry long time ago. Hence, ∇(···) is the classical gradient, while ∇¯(⋅⋅⋅) is weighted by curvatures of curved surfaces and has not appeared in classical geometry. Hence, ∇¯(⋅⋅⋅) is a new gradient operator. Because curvatures depict the local shape of a curved space, ∇¯(⋅⋅⋅) is called the shape gradient operator.

The classical gradient is one of the most fundamental concepts in geometry. Its geometric meaning is very clear: the fastest changing direction and the largest changing rate of a field function. Researchers in mechanics are quite familiar with the classical gradient, but they inquire the shape gradient once it is published: “What is the geometric meaning of the shape gradient?”

Ten years ago, the author had asked the same question at the moment when ∇¯(⋅⋅⋅) was defined. Unfortunately, he has no answer as yet. The formulation of the definition for ∇¯(⋅⋅⋅) is very clear and the geometric invariant properties of ∇¯(⋅⋅⋅) are very beautiful [3, 4], but the geometric meaning of ∇¯(⋅⋅⋅) remains unclear. This situation is very strange and we are very disappointed.

Various researchers’ viewpoints also diverge. One viewpoint is as follows: “If the geometric meaning is not clear, then the concept has no value not only in geometry but also in mechanics”. Another viewpoint is as follows: “Now that 〚∇, ∇¯〛 has symmetric geometric properties [3, 4], then just as ∇(···), ∇¯(⋅⋅⋅) should also be useful”.

2.4 Physical meaning

“What is the physical meaning of ∇¯(⋅⋅⋅)?” The author’s answer is “shape gradient driving force”. In common sense, every term in the equilibrium differential equation represents a force. In Eq. (1), f=f(H, K, λ, Δp) is a generalized force, and ∇⋅(∇ϕ,H/2+∇¯ϕ,K) is certainly a force.

Now we make a comparison with the balance law of linear momentum in continuum mechanics:

Here ∇(⋅⋅⋅)=∂(⋅⋅⋅)∂xi+∂(⋅⋅⋅)∂yj+∂(⋅⋅⋅)∂zk is the classical gradient (divergence operator) in Euclidean space. If Eqs. (1) and (4) are compared, then a correspondence between ∇⋅(∇ϕ,H/2+∇¯ϕ,K) and ∇·σ can be deduced. The quantity σ is the stress or the internal force per unit area. Accordingly, (∇ϕ,H/2+∇¯ϕ,K) may also be viewed as an internal force.

Next, we recall the physical meaning of the classical gradient. The classical gradient is one of the most important concepts in mechanics. It is well known that the gradient of a potential is a force. In electricity, the gradient of the electric potential is the electric field force. In fluid mechanics, the gradient of pressure is the force driving flows. In thermodynamics, the gradient of temperature is the force leading to heat conduction. In diffusion, the gradient of concentration is the force driving matter transport. In the subterm (∇ϕ,H/2+∇¯ϕ,K),ϕ_,H is a potential and ∇ϕ_,H is its gradient or a driving force. If ∇ϕ_,H is a driving force, then ∇¯ϕ,K is also a driving force. We call ∇(···) the classical gradient driving force, and ∇¯(⋅⋅⋅) the shape gradient driving force.

Of course, the “shape gradient driving force” is the driving force induced by changes in geometric shapes. It seems to be a geometrical force instead of physical force. Obviously, this is an abnormal driving force. This explanation has been seriously criticized in recent years. Statements such as “the interpretation is rather unconvincing”, “the shape gradient driving force lacks experimental basis”, “it is not serious to annotate an abstract symbol ∇¯(⋅⋅⋅) as a driving force”, and “this is just a term of comparison and judgment, but is far from a physical existence” have been used to question its usefulness. For this reason, we will call it an abnormal force.

2.5 Inspirations from shallow shell mechanics

At the beginning, micro/nanomechanics has been regarded as the only platform for considering ∇¯(⋅⋅⋅). However, this viewpoint is soon overturned: ∇¯(⋅⋅⋅) appears not only in micro/nanomechanics but also in shallow shell mechanics. Shallow shell theory can be expressed by the classical Donnell equations. After careful examination, we find that the Donnell equations can finally be written as

To one’s surprise, ∇¯(⋅⋅⋅) appeared in the Donnell equations 70 years ago but no sufficient attention to it was paid. Here, D is the bending stiffness of the shell, E is Young’s modulus, 2h is the thickness, and q is the external force per unit area along the normal direction of the shell. Equation (5a) is the equilibrium differential equation along the normal direction. Equation (5b) is the deformation compatibility equation.

Equation (5a) can be interpreted as follows: if the shell is shallow enough, then ∇¯c→0, and Eq. (5a) will degenerate to the equilibrium differential equation for an elastic thin plate, i.e., ∇·[D∇(∇·∇w)]=q. For a thin plate, the normal internal force ∇·[D∇(∇·∇w)] is balanced by the normal external force q. But for a shallow shell, ∇·[D∇(∇·∇w)] is only one part of the internal forces. Another part is induced by the space bending, i.e., ∇⋅∇¯c. The two parts of the internal forces have to work together to balance the normal external force q.

Now that ∇·[D∇(∇·∇w)] and ∇⋅∇¯c are all normal internal forces, and D∇(∇·∇w) is the driving force in the curved surface, it follows that ∇¯c is certainly a driving force in the curved surface. Obviously, the shape gradient ∇¯(⋅⋅⋅) not only cooperates with the classical gradient ∇(···) but also plays the roles of an abnormal driving force.

In comparison with ∇¯ϕ,K in Eq. (1), ∇¯c in Eq. (5a) is more like a shape gradient driving force. When we say that ∇¯ϕ,K is a shape gradient driving force, its meaning may be a little bit unclear, because curvatures that determine the shape exist not only in ∇¯(⋅⋅⋅) but also in ϕ_,K. But ∇¯c is different, because the stress function c does not depend on curvature. Hence, when we say that ∇¯c is a shape gradient driving force, the shape means exactly the curvatures in ∇¯(⋅⋅⋅).

Thin shell mechanics is a mature branch of classical mechanics. Its concepts and theories have been well accepted. Even so, it does not seem, however, that the mechanism has fully appreciated that ∇¯c is indeed a shape gradient driving force.

3.1 Driving forces induced by curvatures

As driving forces, ∇(···) and ∇¯(⋅⋅⋅) are different. ∇(···) is a familiar driving force. It can be seen either in a flat space or in a curved space. The unfamiliar driving force ∇¯(⋅⋅⋅) seems to exist only in a curved space. In a flat surface, ∇¯(⋅⋅⋅)≡0. If the flat surface is bent into a curved one, then it suddenly appears like a ghost. In a plate, there is no ∇¯(⋅⋅⋅). In a shallow shell, it jumps out. We have never seen it in mechanics. In fact, we have never been conscious of its existence. Thus, we state Proposition I:

Curved space may induce a shape gradient driving force.

Or

Curvatures may induce a shape gradient driving force.

Proposition I deviates from traditional concepts. This is not strange. Because the shape gradient is a subversive concept, the corresponding proposition is certainly a subversive proposition.

3.2 Driving forces induced by gradients of curvatures

Classical gradient driving force is a conventional concept, and a corresponding proposition may also be a conventional one. Nevertheless, this idea is not valid in micro/nano curved spaces.

Let us look at Eq. (1) again. Either for the classical gradient driving force ∇ϕ_,H or for the shape gradient driving force ∇¯ϕ,K, we have

Here ϕ_,HH=∂²ϕ/∂H², ϕ_,HK=ϕ_,KH=∂²ϕ/∂H∂K, and ϕ_,KK=∂²ϕ/∂K² are all functions of curvatures. The quantities ∇H and ∇¯H are respectively the classical gradient and the shape gradient of the mean curvature, whereas ∇K and ∇¯K are respectively the classical gradient and shape gradient of the Gauss curvature. How do we understand Eq. (6)?

Equation (6)₂ strengthens Proposition I: curved space induces the shape gradient driving force ∇¯ϕ,K. In contrast, Eq. (6)₁ is beyond our current interpretations: curved space can also induce a classical gradient driving force ∇ϕ_,H. Then we are led to Proposition II:

Curved space may induce driving forces.

Here the driving forces include both the shape gradient driving force and the classical gradient driving force.

The driving forces in Proposition II have internal analytical structures. In the analytical structures of ∇ϕ_,H and ∇¯ϕ,K, two basic parts are included: one is the function terms of curvatures, i.e., ϕ_,HH, ϕ_,HK, ϕ_,KH, ϕ_,KK and another is the gradient terms of curvatures, i.e., ∇H, ∇K, ∇¯H,∇¯K. Among them, ϕ_,HH, ϕ_,HK weight linearly the classical gradient terms ∇H and ∇K, and ϕ_,KH, ϕ_,KK weight linearly the shape gradient terms ∇¯H and ∇¯K. This leads to Proposition III:

Curvatures and the gradients of curvatures form the essential factors of driving forces.

Curvatures depict the local bending extent of matter spaces, and the gradients of curvatures describe the non-uniform bending extent of matter spaces. To determine the driving forces on micro/nano curved surfaces, curvatures alone are not enough; the gradients of curvatures also need to be considered as they are indispensable. Propositions II and III are also subversive. It follows that in micro/nano curved spaces, both the classical gradient driving force and the shape gradient driving force are abnormal driving forces.

4.1 Strategies for proof

The first strategy is to select two “universal” physical configurations [13–17] as a precondition for the proof. One is the short-range interaction between a pair of particles, and another is the highly curved matter space form. The reasons are as follows. On the one hand, it is natural to select the interaction between a pair of particles as a physical base configuration to begin with. On the other hand, both the abnormal driving forces and the subversive propositions are concerned with micro/nano curved spaces. Hence, it is reasonable to take the curved space form as another geometric-based configuration.

The second strategy is to select the interaction potential as the starting point of the proof. According to the viewpoint of mathematicians, “the concept of potential is a mathematical inspiration, which enables us to solve a physical problem which cannot be approached without such a concept”.

The third strategy is to abstract fundamental interaction modes as the carrier for the proof. Such modes should be as universal as possible to ensure the generality of the result. Besides, such modes should be as simple as possible to ensure the transparency of the process.

4.2 Unified pair potential

At micro/nanoscale, short-range interaction potentials universally exist between particles. Among them, the van der Waals pair potential, i.e., u(r)=u12-u6=4ε[(σr)12-(σr)6], is very popular. Various short-range potentials can be unified by the power form

Such potentials control the direction of biological evolution and are of essential importance in micro/nanobiomechanics.

4.3 Abstracted fundamental modes

A curved matter surface is usually discontinuous, but it may be abstracted as a continuous and smooth curved surface. Certainly, continuous and smooth curved surfaces do not exist in nature, but this approximation does not diminish the effectiveness of employing the concept of the abstract modes. This judgment may be supported by many examples. The highly curved bilayer molecule membrane with nanothickness (Figure 7) is taken as the background for the abstracted modes. The interaction potential between a molecule pair is chosen as Eq. (5). In this connection, the question may be asked: “How do we derive the interaction potential of the whole bilayer molecule membrane?”

Figure 7

The nano bilayer molecule membrane.

One conventional way is to sum up the potentials of all molecule pairs. This scheme is workable but limited. (a) The geometric basis is Euclidean and cannot reflect the geometric feature of the curved space form. (b) It is effective just case by case, and has no universality.

To abstract fundamental modes from the bilayer molecule membrane, the following basic idea is employed. (a) The bilayer molecule membrane is abstracted as two continuous and smooth curved surfaces: S_in and S_out. All molecules are abstracted as particles arranged inside S_in and S_out. (b) The particle number density ρ_s, i.e., the number of particles per unit area, is defined on the basis of the area measure on the curved surface. (c) Just one particle p inside S_in is targeted. The interactions between a particle p and all other particles located inside S_in and S_out are considered.

Particle p should be paired with other particles inside S_in and S_out to construct the interaction potential. There are different ways to pair these particles. Two of them are extremely special, from which two fundamental modes with universality can be abstracted [16, 17]: one is the interactions between particle p and all other particles inside S_in. This is because particle p itself is also located inside S_in, from which we may induce the “inner particle/curved surface” interaction mode (Figure 8). Another way is to consider the interactions between particle p and all particles inside S_out. This is because particle p is located outside S_out, from which we may induce the “outer particle/curved surface” interaction mode (Figure 9).

Figure 8

The “inner particle/curved surface” interaction mode.

Figure 9

The “outer particle/curved surface” interaction mode.

One may think that these modes, like toys, are “too simple and useless”. However, the toy-like modes are of excellent characteristics. First, they are extremely simple, i.e., only one particle p is included. Second, they show the unique feature, i.e., the fundamental characteristic of the curved space form is preserved to the extreme extent. Third, they have universality, i.e., other complicated interactions can be regarded as the summation of the two fundamental modes.

In fact, the abstract idea hidden behind the above fundamental modes is rooted in the Newtonian tradition. Newton treated the sun and the earth as two particles when he studied gravity. To reveal the influence of the internal structure on gravity, he again treated the sun and the earth as a sphere and a particle, respectively.

4.4 Curvature-based potential of a particle/curved surface

Pair particles interact along a straight line. However, once such a straight line interaction is controlled by the highly curved space form in the fundamental modes, new effects are created. In the “inner particle/curved surface” mode, the potential of particle p is

Here Sin∗ is the curved surface with particle p excluded. In the “outer particle/curved surface” mode, the potential of particle p is

It should be noted that the number density ρ_s is the key point to transform a discrete problem into a continuous problem.

As surface integrals, Un(in) and Un(out) reflect the properties of the global structures. However, the highly local feature of the pair potential u_n(r) ensures that interactions mainly take place between particle p and the nearest particles on curved surfaces. Because the distribution of the nearest particles is controlled by the local shape of the curved surface, and the local shape is depicted by the curvature invariables, finally both Un(in) and Un(out) are determined by curvatures and can be expressed as the functions of curvatures.

For the “inner particle/curved surface” mode, we have [16–18]

Here τ is the characteristic radius of particle p, H and K are, respectively, the mean curvature and Gauss curvature of S_in at point p, while U¯n(in)=Un(in)|H=0K=0 is the interaction potential between particle p and the tangential plane of S_in at point p.

For the “outer particle/curved surface” mode, we have [16]

Here h is the shortest distance from particle p to the nearest point O on S_out, H and K are, respectively, the mean curvature and Gauss curvature of S_out at the nearest point O, while U¯n(out)=Un(out)|H=0K=0 is the interaction potential between particle p and the tangential plane of S_out at point O.

In the two fundamental modes, the potentials of particle p are all expressed as functions of curvatures. The curvature-based potentials can be generally written as Un(in)=Un(in)(H, K, τ) and Un(out)=Un(out)(H, K, h).

4.5 Classical gradient driving force acted on a particle

We assume the following idealized movement. Curved surfaces are rigid, i.e., the shapes are unchangeable, and the location of particle p is perturbed. This is the case of interaction between particle p and a hard curved surface.

In the “inner particle/curved surface” mode, particle p will feel a driving force along the tangential plane of S_in when it moves along S_in:

Here ∇(···) is the classical gradient defined in S_in.

In the “outer particle/curved surface” mode, particle p will feel driving forces:

Here ∇(···) is the classical gradient defined in the parallel curved surface with h=Const., Ft(out) is the driving force along the tangential plane of the parallel curved surface, and Fz(out) is the driving force along the normal direction.

This is a conceptual type of constructive proof for the classical gradient driving force. The classical gradient driving force has indeed a clear physical meaning, although it is abnormal.

4.6 Shape gradient driving force acted on a particle

We assume the following idealized movement: curved surfaces are flexible, i.e., the shape is pertubated, which induces the variation of the driving forces acted on particle p. This is the case of the interaction between particle p and a soft curved surface.

For simplicity, here we do not seek a conceptual constructive proof. Instead, we provide the following “existence” proof.

The interaction potentials between particle p and the soft curved surfaces S_in and S_out can be unified as U_n=U_n(H, K). The shape pertubation of the soft curved surface can be taken as the normal fluctuation δw, which is expected to cause the following “chain reactions”.

First, δw will lead to the variations in curvatures [9–11], i.e., δw→H→H+δH, K→K+δK:

Second, the changes in curvatures will cause corresponding variations in the potential of particle p, i.e., H+δH, K+δK→U_n+δU_n:

Finally, the change in the potential of particle p will lead to the variations in driving forces, i.e., U_n+δU_n→F→F+δF.

In the variation of driving forces δF, both the classical gradient driving force ∇(···) and the shape gradient driving force ∇¯(⋅⋅⋅) are included. This may be viewed as an “existence” proof for the shape gradient driving force ∇¯(⋅⋅⋅).

4.7 Soft/hard surfaces and nests for driving forces

At the micro/nanoscale, although both the classical gradient driving force ∇(···) and the shape gradient driving force ∇¯(⋅⋅⋅) are viewed as physical quantities, their nests are different. ∇(···) lives everywhere. It exists not only on hard curved surfaces but also on soft curved surfaces. ∇¯(⋅⋅⋅) is different. It seldom occurs on hard curved surfaces, but always “hides” in soft curved surfaces. In comparison, the driving force ∇(···) is more active and easier to measure, whereas the driving force ∇¯(⋅⋅⋅) comes and goes like a ghost and is difficult to capture. The driving force ∇(···) controls the movement of matters inside/outside the soft/hard curved surfaces in an eye-catching way, whereas the driving force ∇¯(⋅⋅⋅) influences the evolution of matter inside/outside the soft/hard curved surfaces in a concealed way.

Both the driving force ∇(···) and the driving force ∇¯(⋅⋅⋅) live in soft curved surfaces. So, in comparison with hard curved surfaces, soft curved surfaces are much more difficult to understand. In other words, the dynamics inside/outside the soft curved surfaces are much more complicated (see Figure 5).

The driving force ∇(···) and the driving force ∇¯(⋅⋅⋅) can be combined into the driving forces not only in the normal direction but also in the tangential direction. They can drive particle p to move both in the normal direction and in the tangential direction. It is expected that the tangential driving force is the key factor for controlling diffusion and pattern evolution on highly curved surfaces. Besides, it may be critical in the explanation of the new phenomena depicted in Figures 1–5.