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99-409: Wetting is important in the bonding or adherence of two materials. Wetting and the surface forces that control wetting are also responsible for other related effects, including capillary effects. Surfactants can be used to increase the wetting power of a liquid like water. Wetting is a focus of research attention in nanotechnology and nanoscience studies due to the advent of many nanomaterials in

198-550: A denotes a constant belonging to some field K of scalars (for example, the real numbers ) and x and y are elements of a vector space , which might be K itself. In other terms the linear function preserves vector addition and scalar multiplication . Some authors use "linear function" only for linear maps that take values in the scalar field; these are more commonly called linear forms . The "linear functions" of calculus qualify as "linear maps" when (and only when) f (0, ..., 0) = 0 , or, equivalently, when

297-490: A triangle , they are constrained by the triangle inequalities, γ ij < γ jk + γ ik meaning that not one of the surface tensions can exceed the sum of the other two. If three fluids with surface energies that do not follow these inequalities are brought into contact, no equilibrium configuration consistent with Figure 3 will exist. If the β phase is replaced by a flat rigid surface, as shown in Figure 5, then β = π, and

396-450: A chemical bond, where the higher the associated electronegativity then the more it attracts electrons. Electronegativity serves as a simple way to quantitatively estimate the bond energy , which characterizes a bond along the continuous scale from covalent to ionic bonding . A large difference in electronegativity leads to more polar (ionic) character in the bond. Ionic bonding is a type of electrostatic interaction between atoms that have

495-494: A covalent bond as an orbital formed by combining the quantum mechanical Schrödinger atomic orbitals which had been hypothesized for electrons in single atoms. The equations for bonding electrons in multi-electron atoms could not be solved to mathematical perfection (i.e., analytically ), but approximations for them still gave many good qualitative predictions and results. Most quantitative calculations in modern quantum chemistry use either valence bond or molecular orbital theory as

594-453: A direction in space, allowing them to be shown as single connecting lines between atoms in drawings, or modeled as sticks between spheres in models. In a polar covalent bond , one or more electrons are unequally shared between two nuclei. Covalent bonds often result in the formation of small collections of better-connected atoms called molecules , which in solids and liquids are bound to other molecules by forces that are often much weaker than

693-527: A heterogeneous surface, the Wenzel model is not sufficient. A more complex model is needed to measure how the apparent contact angle changes when various materials are involved. This heterogeneous surface, like that seen in Figure 8, is explained using the Cassie–Baxter equation ( Cassie's law ): Here the r f is the roughness ratio of the wet surface area and f is the fraction of solid surface area wet by

792-423: A homogeneous surface. The roughness ratio is defined as the ratio of true area of the solid surface to the apparent area. θ is the contact angle for a system in thermodynamic equilibrium, defined for a perfectly flat surface. Although Wenzel's equation demonstrates the contact angle of a rough surface is different from the intrinsic contact angle, it does not describe contact angle hysteresis . When dealing with

891-477: A large electronegativity difference. There is no precise value that distinguishes ionic from covalent bonding, but an electronegativity difference of over 1.7 is likely to be ionic while a difference of less than 1.7 is likely to be covalent. Ionic bonding leads to separate positive and negative ions . Ionic charges are commonly between −3 e to +3 e . Ionic bonding commonly occurs in metal salts such as sodium chloride (table salt). A typical feature of ionic bonds

990-541: A lone pair that can be shared is described as an electron-pair donor or Lewis base . The electrons are shared roughly equally between the atoms in contrast to ionic bonding. Such bonding is shown by an arrow pointing to the Lewis acid. (In the Figure, solid lines are bonds in the plane of the diagram, wedged bonds point towards the observer, and dashed bonds point away from the observer.) Transition metal complexes are generally bound by coordinate covalent bonds. For example,

1089-400: A model of the chemical bond in 1913. According to his model for a diatomic molecule , the electrons of the atoms of the molecule form a rotating ring whose plane is perpendicular to the axis of the molecule and equidistant from the atomic nuclei. The dynamic equilibrium of the molecular system is achieved through the balance of forces between the forces of attraction of nuclei to the plane of

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1188-513: A net positive charge, and the other to assume a net negative charge. The bond then results from electrostatic attraction between the positive and negatively charged ions . Ionic bonds may be seen as extreme examples of polarization in covalent bonds. Often, such bonds have no particular orientation in space, since they result from equal electrostatic attraction of each ion to all ions around them. Ionic bonds are strong (and thus ionic substances require high temperatures to melt) but also brittle, since

1287-400: A part of the shell of two different atoms and cannot be said to belong to either one exclusively." Also in 1916, Walther Kossel put forward a theory similar to Lewis' only his model assumed complete transfers of electrons between atoms, and was thus a model of ionic bonding . Both Lewis and Kossel structured their bonding models on that of Abegg's rule (1904). Niels Bohr also proposed

1386-442: A permanent dipole in one molecule and an induced dipole in another molecule. Hydrogen bonds of the form A--H•••B occur when A and B are two highly electronegative atoms (usually N, O or F) such that A forms a highly polar covalent bond with H so that H has a partial positive charge, and B has a lone pair of electrons which is attracted to this partial positive charge and forms a hydrogen bond. Hydrogen bonds are responsible for

1485-418: A phase boundary, involving advancing and receding contact angles, is known as dynamic wetting. The difference between dynamic and static wetting angles is proportional to the capillary number , C a {\displaystyle Ca} , When a contact line advances, covering more of the surface with liquid, the contact angle is increased and is generally related to the velocity of the contact line. If

1584-441: A rough texture. The rough texture of a surface can fall into one of two categories: homogeneous or heterogeneous. A homogeneous wetting regime is where the liquid fills in the grooves of a rough surface. A heterogeneous wetting regime, though, is where the surface is a composite of two types of patches. An important example of such a composite surface is one composed of patches of both air and solid. Such surfaces have varied effects on

1683-709: A sessile droplet to the bulk thermodynamics, the energy at the three phase contact boundary, and the curvature of the surface α. For the special case of a sessile droplet on a flat surface (α=0), The first two terms are the modified Young's equation, while the third term is due to the Laplace pressure. This nonlinear equation correctly predicts the sign and magnitude of κ, the flattening of the contact angle at very small scales, and contact angle hysteresis. For many surface/adsorbate configurations, surface energy data and experimental observations are unavailable. As wetting interactions are of great importance in various applications, it

1782-451: A sodium cyanide crystal. When such crystals are melted into liquids, the ionic bonds are broken first because they are non-directional and allow the charged species to move freely. Similarly, when such salts dissolve into water, the ionic bonds are typically broken by the interaction with water but the covalent bonds continue to hold. For example, in solution, the cyanide ions, still bound together as single CN ions, move independently through

1881-547: A solid has to do with the bulk nature of the solid itself. Solids such as metals, glasses , and ceramics are known as 'hard solids' because the chemical bonds that hold them together (e.g., covalent , ionic , or metallic ) are very strong. Thus, it takes a large amount of energy to break these solids (alternatively, a large amount of energy is required to cut the bulk and make two separate surfaces), so they are termed "high-energy". Most molecular liquids achieve complete wetting with high-energy surfaces. The other type of solid

1980-402: A starting point, although a third approach, density functional theory , has become increasingly popular in recent years. In 1933, H. H. James and A. S. Coolidge carried out a calculation on the dihydrogen molecule that, unlike all previous calculation which used functions only of the distance of the electron from the atomic nucleus, used functions which also explicitly added the distance between

2079-413: A surface with straight chains. Lower critical surface tension means a less wettable material surface. An ideal surface is flat, rigid, perfectly smooth, chemically homogeneous, and has zero contact angle hysteresis . Zero hysteresis implies the advancing and receding contact angles are equal. In other words, only one thermodynamically stable contact angle exists. When a drop of liquid is placed on such

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2178-441: A surface, the characteristic contact angle is formed as depicted in Figure 1. Furthermore, on an ideal surface, the drop will return to its original shape if it is disturbed. The following derivations apply only to ideal solid surfaces; they are only valid for the state in which the interfaces are not moving and the phase boundary line exists in equilibrium. Figure 3 shows the line of contact where three phases meet. In equilibrium ,

2277-421: A vacancy which allows the addition of one or more electrons. These newly added electrons potentially occupy a lower energy-state (effectively closer to more nuclear charge) than they experience in a different atom. Thus, one nucleus offers a more tightly bound position to an electron than does another nucleus, with the result that one atom may transfer an electron to the other. This transfer causes one atom to assume

2376-444: Is a characteristic of only the solid. Knowing the critical surface tension of a solid, it is possible to predict the wettability of the surface. The wettability of a surface is determined by the outermost chemical groups of the solid. Differences in wettability between surfaces that are similar in structure are due to differences in the packing of the atoms. For instance, if a surface has branched chains, it will have poorer packing than

2475-434: Is a common type of bonding in which two or more atoms share valence electrons more or less equally. The simplest and most common type is a single bond in which two atoms share two electrons. Other types include the double bond , the triple bond , one- and three-electron bonds , the three-center two-electron bond and three-center four-electron bond . In non-polar covalent bonds, the electronegativity difference between

2574-453: Is a covalent bond in which the two shared bonding electrons are from the same one of the atoms involved in the bond. For example, boron trifluoride (BF 3 ) and ammonia (NH 3 ) form an adduct or coordination complex F 3 B←NH 3 with a B–N bond in which a lone pair of electrons on N is shared with an empty atomic orbital on B. BF 3 with an empty orbital is described as an electron pair acceptor or Lewis acid , while NH 3 with

2673-441: Is a covalent bond with a significant ionic character . This means that the two shared electrons are closer to one of the atoms than the other, creating an imbalance of charge. Such bonds occur between two atoms with moderately different electronegativities and give rise to dipole–dipole interactions . The electronegativity difference between the two atoms in these bonds is 0.3 to 1.7. A single bond between two atoms corresponds to

2772-453: Is a situation unlike that in covalent crystals, where covalent bonds between specific atoms are still discernible from the shorter distances between them, as measured via such techniques as X-ray diffraction . Ionic crystals may contain a mixture of covalent and ionic species, as for example salts of complex acids such as sodium cyanide , NaCN. X-ray diffraction shows that in NaCN, for example,

2871-570: Is because DFT calculations are generally conducted assuming conditions of zero thermal movement of atoms, essentially meaning the simulation is conducted at absolute zero . This simplification nevertheless yields results that are relevant for the adsorption of water under realistic conditions and the use of ice for the theoretical simulation of wetting is commonplace. Unlike ideal surfaces, real surfaces do not have perfect smoothness, rigidity, or chemical homogeneity. Such deviations from ideality result in phenomenon called contact angle hysteresis , which

2970-542: Is computed to be: Now, we recall that the boundary is free in the x {\displaystyle x} direction and L {\displaystyle L} is a free parameter. Therefore, we must have: At the boundary y ( L ) = 0 {\displaystyle y(L)=0} and ( 1 + y ′ 2 ) − 1 / 2 = cos ⁡ θ {\displaystyle (1+y'^{2})^{-1/2}=\cos \theta } , therefore we recover

3069-405: Is consistent with the geometrical restriction that α + β + θ = 2 π {\displaystyle \alpha +\beta +\theta =2\pi } , and applying the law of sines and law of cosines to it produce relations that describe how the interfacial angles depend on the ratios of surface energies. Because these three surface energies form the sides of

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3168-439: Is defined as the difference between the advancing (θ a ) and receding (θ r ) contact angles When the contact angle is between the advancing and receding cases, the contact line is considered to be pinned and hysteretic behaviour can be observed, namely contact angle hysteresis . When these values are exceeded, the displacement of the contact line, such as the one in Figure 3, will take place by either expansion or retraction of

3267-405: Is determined by the balance between adhesive and cohesive forces. As the tendency of a drop to spread out over a flat, solid surface increases, the contact angle decreases. Thus, the contact angle provides an inverse measure of wettability. A contact angle less than 90° (low contact angle) usually indicates that wetting of the surface is very favorable, and the fluid will spread over a large area of

3366-442: Is discussed. Sometimes, even the non-bonding valence shell electrons (with the two-dimensional approximate directions) are marked, e.g. for elemental carbon . C . Some chemists may also mark the respective orbitals, e.g. the hypothetical ethene anion ( \ C=C / ) indicating the possibility of bond formation. Strong chemical bonds are the intramolecular forces that hold atoms together in molecules . A strong chemical bond

3465-413: Is formed from the transfer or sharing of electrons between atomic centers and relies on the electrostatic attraction between the protons in nuclei and the electrons in the orbitals. The types of strong bond differ due to the difference in electronegativity of the constituent elements. Electronegativity is the tendency for an atom of a given chemical element to attract shared electrons when forming

3564-465: Is free (by virtue of its wave nature ) to be associated with a great many atoms at once. The bond results because the metal atoms become somewhat positively charged due to loss of their electrons while the electrons remain attracted to many atoms, without being part of any given atom. Metallic bonding may be seen as an extreme example of delocalization of electrons over a large system of covalent bonds, in which every atom participates. This type of bonding

3663-448: Is no clear line to be drawn between them. However it remains useful and customary to differentiate between different types of bond, which result in different properties of condensed matter . In the simplest view of a covalent bond , one or more electrons (often a pair of electrons) are drawn into the space between the two atomic nuclei. Energy is released by bond formation. This is not as a result of reduction in potential energy, because

3762-426: Is often desired to predict and compare the wetting behavior of various material surfaces with particular crystallographic orientations, with relation to water or other adsorbates. This can be done from an atomistic perspective with tools including molecular dynamics and density functional theory . In the theoretical prediction of wetting by ab initio approaches such as DFT, ice is commonly substituted for water. This

3861-574: Is often very strong (resulting in the tensile strength of metals). However, metallic bonding is more collective in nature than other types, and so they allow metal crystals to more easily deform, because they are composed of atoms attracted to each other, but not in any particularly-oriented ways. This results in the malleability of metals. The cloud of electrons in metallic bonding causes the characteristically good electrical and thermal conductivity of metals, and also their shiny lustre that reflects most frequencies of white light. Early speculations about

3960-472: Is sometimes concerned only with the functional group of the molecule. Thus, the molecular formula of ethanol may be written in conformational form, three-dimensional form, full two-dimensional form (indicating every bond with no three-dimensional directions), compressed two-dimensional form (CH 3 –CH 2 –OH), by separating the functional group from another part of the molecule (C 2 H 5 OH), or by its atomic constituents (C 2 H 6 O), according to what

4059-455: Is that the species form into ionic crystals, in which no ion is specifically paired with any single other ion in a specific directional bond. Rather, each species of ion is surrounded by ions of the opposite charge, and the spacing between it and each of the oppositely charged ions near it is the same for all surrounding atoms of the same type. It is thus no longer possible to associate an ion with any specific other single ionized atom near it. This

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4158-449: Is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form . In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps . In linear algebra, a linear function is a map f between two vector spaces such that Here

4257-528: Is weak molecular crystals (e.g., fluorocarbons , hydrocarbons , etc.) where the molecules are held together essentially by physical forces (e.g., van der Waals forces and hydrogen bonds ). Since these solids are held together by weak forces, a very low amount of energy is required to break them, thus they are termed "low-energy". Depending on the type of liquid chosen, low-energy surfaces can permit either complete or partial wetting. Dynamic surfaces have been reported that undergo changes in surface energy upon

4356-778: The London dispersion force , and hydrogen bonding . Since opposite electric charges attract, the negatively charged electrons surrounding the nucleus and the positively charged protons within a nucleus attract each other. Electrons shared between two nuclei will be attracted to both of them. "Constructive quantum mechanical wavefunction interference " stabilizes the paired nuclei (see Theories of chemical bonding ). Bonded nuclei maintain an optimal distance (the bond distance) balancing attractive and repulsive effects explained quantitatively by quantum theory . The atoms in molecules , crystals , metals and other forms of matter are held together by chemical bonds, which determine

4455-433: The atom in which a positively charged center is surrounded by a number of revolving electrons, in the manner of Saturn and its rings. Nagaoka's model made two predictions: Rutherford mentions Nagaoka's model in his 1911 paper in which the atomic nucleus is proposed. At the 1911 Solvay Conference, in the discussion of what could regulate energy differences between atoms, Max Planck stated: "The intermediaries could be

4554-405: The electrostatic force between oppositely charged ions as in ionic bonds or through the sharing of electrons as in covalent bonds , or some combination of these effects. Chemical bonds are described as having different strengths: there are "strong bonds" or "primary bonds" such as covalent , ionic and metallic bonds, and "weak bonds" or "secondary bonds" such as dipole–dipole interactions ,

4653-428: The linear combination of atomic orbitals and ligand field theory . Electrostatics are used to describe bond polarities and the effects they have on chemical substances. A chemical bond is an attraction between atoms. This attraction may be seen as the result of different behaviors of the outermost or valence electrons of atoms. These behaviors merge into each other seamlessly in various circumstances, so that there

4752-481: The melting point ) of a substance. Van der Waals forces are interactions between closed-shell molecules. They include both Coulombic interactions between partial charges in polar molecules, and Pauli repulsions between closed electrons shells. Keesom forces are the forces between the permanent dipoles of two polar molecules. London dispersion forces are the forces between induced dipoles of different molecules. There can also be an interaction between

4851-440: The silicate minerals in many types of rock) then the structures that result may be both strong and tough, at least in the direction oriented correctly with networks of covalent bonds. Also, the melting points of such covalent polymers and networks increase greatly. In a simplified view of an ionic bond , the bonding electron is not shared at all, but transferred. In this type of bond, the outer atomic orbital of one atom has

4950-427: The surface tension (γ LV ) of the liquid decreased. Thus, he was able to establish a linear function between cos θ and the surface tension (γ LV ) for various organic liquids. A surface is more wettable when γ LV and θ is low. Zisman termed the intercept of these lines when cos θ = 1 as the critical surface tension (γ c ) of that surface. This critical surface tension is an important parameter because it

5049-465: The Danish physicist Øyvind Burrau . This work showed that the quantum approach to chemical bonds could be fundamentally and quantitatively correct, but the mathematical methods used could not be extended to molecules containing more than one electron. A more practical, albeit less quantitative, approach was put forward in the same year by Walter Heitler and Fritz London . The Heitler–London method forms

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5148-464: The Young equation. The Young equation assumes a perfectly flat and rigid surface often referred to as an ideal surface . In many cases, surfaces are far from this ideal situation, and two are considered here: the case of rough surfaces and the case of smooth surfaces that are still real (finitely rigid). Even in a perfectly smooth surface, a drop will assume a wide spectrum of contact angles ranging from

5247-468: The Young–Dupré equation is an indicator that there is no equilibrium configuration with a contact angle between 0 and 180° for those situations. A useful parameter for gauging wetting is the spreading parameter S , When S > 0, the liquid wets the surface completely (complete wetting). When S < 0, partial wetting occurs. Combining the spreading parameter definition with the Young relation yields

5346-405: The Young–Dupré equation: which only has physical solutions for θ when S < 0. With improvements in measuring techniques such as AFM, confocal microscopy and SEM, researchers were able to produce and image droplets at ever smaller scales. With the reduction in droplet size came new experimental observations of wetting. These observations confirm that the modified Young's equation does not hold at

5445-437: The application of an appropriate stimuli. For example, a surface presenting photon-driven molecular motors was shown to undergo changes in water contact angle when switched between bistable conformations of differing surface energies. Low-energy surfaces primarily interact with liquids through dispersive ( van der Waals ) forces. William Zisman produced several key findings: Zisman observed that cos θ increases linearly as

5544-491: The approximations differ, and one approach may be better suited for computations involving a particular system or property than the other. Unlike the spherically symmetrical Coulombic forces in pure ionic bonds, covalent bonds are generally directed and anisotropic . These are often classified based on their symmetry with respect to a molecular plane as sigma bonds and pi bonds . In the general case, atoms form bonds that are intermediate between ionic and covalent, depending on

5643-487: The atoms involved can be understood using concepts such as oxidation number , formal charge , and electronegativity . The electron density within a bond is not assigned to individual atoms, but is instead delocalized between atoms. In valence bond theory, bonding is conceptualized as being built up from electron pairs that are localized and shared by two atoms via the overlap of atomic orbitals. The concepts of orbital hybridization and resonance augment this basic notion of

5742-480: The attraction of the two electrons to the two protons is offset by the electron-electron and proton-proton repulsions. Instead, the release of energy (and hence stability of the bond) arises from the reduction in kinetic energy due to the electrons being in a more spatially distributed (i.e. longer de Broglie wavelength ) orbital compared with each electron being confined closer to its respective nucleus. These bonds exist between two particular identifiable atoms and have

5841-401: The basis of what is now called valence bond theory . In 1929, the linear combination of atomic orbitals molecular orbital method (LCAO) approximation was introduced by Sir John Lennard-Jones , who also suggested methods to derive electronic structures of molecules of F 2 ( fluorine ) and O 2 ( oxygen ) molecules, from basic quantum principles. This molecular orbital theory represented

5940-498: The bonded atoms is small, typically 0 to 0.3. Bonds within most organic compounds are described as covalent. The figure shows methane (CH 4 ), in which each hydrogen forms a covalent bond with the carbon. See sigma bonds and pi bonds for LCAO descriptions of such bonding. Molecules that are formed primarily from non-polar covalent bonds are often immiscible in water or other polar solvents , but much more soluble in non-polar solvents such as hexane . A polar covalent bond

6039-399: The bonds between sodium cations (Na ) and the cyanide anions (CN ) are ionic , with no sodium ion associated with any particular cyanide . However, the bonds between the carbon (C) and nitrogen (N) atoms in cyanide are of the covalent type, so that each carbon is strongly bound to just one nitrogen, to which it is physically much closer than it is to other carbons or nitrogens in

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6138-520: The constraints is therefore where λ i {\displaystyle \lambda _{i}} are Lagrange multipliers. By definition, the momentum p = ∂ y ′ L {\displaystyle p=\partial _{y'}{\cal {L}}} and the Hamiltonian H = p y ′ − L {\displaystyle {\cal {H}}=py'-{\cal {L}}} which

6237-416: The contact angles of wetting liquids. Cassie–Baxter and Wenzel are the two main models that attempt to describe the wetting of textured surfaces. However, these equations only apply when the drop size is sufficiently large compared with the surface roughness scale. When the droplet size is comparable to that of the underlying pillars, the effect of line tension should be considered. The Wenzel model describes

6336-540: The covalent bonds that hold the molecules internally together. Such weak intermolecular bonds give organic molecular substances, such as waxes and oils, their soft bulk character, and their low melting points (in liquids, molecules must cease most structured or oriented contact with each other). When covalent bonds link long chains of atoms in large molecules, however (as in polymers such as nylon ), or when covalent bonds extend in networks through solids that are not composed of discrete molecules (such as diamond or quartz or

6435-469: The density of two non-interacting H atoms. A double bond has two shared pairs of electrons, one in a sigma bond and one in a pi bond with electron density concentrated on two opposite sides of the internuclear axis. A triple bond consists of three shared electron pairs, forming one sigma and two pi bonds. An example is nitrogen. Quadruple and higher bonds are very rare and occur only between certain transition metal atoms. A coordinate covalent bond

6534-485: The difference between the maximum and minimum valencies of an element is often eight. At this point, valency was still an empirical number based only on chemical properties. However the nature of the atom became clearer with Ernest Rutherford 's 1911 discovery that of an atomic nucleus surrounded by electrons in which he quoted Nagaoka rejected Thomson's model on the grounds that opposite charges are impenetrable. In 1904, Nagaoka proposed an alternative planetary model of

6633-422: The droplet. Figure 6 depicts the advancing and receding contact angles. The advancing contact angle is the maximum stable angle, whereas the receding contact angle is the minimum stable angle. Contact angle hysteresis occurs because many different thermodynamically stable contact angles are found on a nonideal solid. These varying thermodynamically stable contact angles are known as metastable states. Such motion of

6732-401: The electron pair bond. In molecular orbital theory, bonding is viewed as being delocalized and apportioned in orbitals that extend throughout the molecule and are adapted to its symmetry properties, typically by considering linear combinations of atomic orbitals (LCAO). Valence bond theory is more chemically intuitive by being spatially localized, allowing attention to be focused on the parts of

6831-445: The electrons." These nuclear models suggested that electrons determine chemical behavior. Next came Niels Bohr 's 1913 model of a nuclear atom with electron orbits. In 1916, chemist Gilbert N. Lewis developed the concept of electron-pair bonds , in which two atoms may share one to six electrons, thus forming the single electron bond , a single bond , a double bond , or a triple bond ; in Lewis's own words, "An electron may form

6930-436: The forces between ions are short-range and do not easily bridge cracks and fractures. This type of bond gives rise to the physical characteristics of crystals of classic mineral salts, such as table salt. A less often mentioned type of bonding is metallic bonding . In this type of bonding, each atom in a metal donates one or more electrons to a "sea" of electrons that reside between many metal atoms. In this sea, each electron

7029-454: The free liquid-vapor boundary is due to the Laplace pressure , which is proportional to the mean curvature of the droplet, and is non zero. Solving the above equation for both convex and concave surfaces yields: Where the constant parameters A, B, and C are defined as: This equation relates the contact angle θ {\displaystyle \theta } , a geometric property of

7128-406: The function is of only one variable , it is of the form where a and b are constants , often real numbers . The graph of such a function of one variable is a nonvertical line. a is frequently referred to as the slope of the line, and b as the intercept. If a > 0 then the gradient is positive and the graph slopes upwards. If a < 0 then the gradient is negative and

7227-399: The graph slopes downwards. For a function f ( x 1 , … , x k ) {\displaystyle f(x_{1},\ldots ,x_{k})} of any finite number of variables, the general formula is and the graph is a hyperplane of dimension k . A constant function is also considered linear in this context, as it is a polynomial of degree zero or

7326-562: The heels of the invention of the voltaic pile , Jöns Jakob Berzelius developed a theory of chemical combination stressing the electronegative and electropositive characters of the combining atoms. By the mid 19th century, Edward Frankland , F.A. Kekulé , A.S. Couper, Alexander Butlerov , and Hermann Kolbe , building on the theory of radicals , developed the theory of valency , originally called "combining power", in which compounds were joined owing to an attraction of positive and negative poles. In 1904, Richard Abegg proposed his rule that

7425-402: The high boiling points of water and ammonia with respect to their heavier analogues. In some cases a similar halogen bond can be formed by a halogen atom located between two electronegative atoms on different molecules. At short distances, repulsive forces between atoms also become important. In the (unrealistic) limit of "pure" ionic bonding , electrons are perfectly localized on one of

7524-427: The homogeneous wetting regime, as seen in Figure 7, and is defined by the following equation for the contact angle on a rough surface: where θ ∗ {\displaystyle \theta ^{*}} is the apparent contact angle which corresponds to the stable equilibrium state (i.e. minimum free energy state for the system). The roughness ratio, r, is a measure of how surface roughness affects

7623-799: The interface as a curve y ( x ) {\displaystyle y(x)} for x ∈ I = [ 0 , L ] {\displaystyle x\in I=[0,L]} where L {\displaystyle L} is a free parameter. The free energy to be minimized is with the constraints y ( 0 ) = y ( L ) = 0 {\displaystyle y(0)=y(L)=0} which we can write as ∫ I y ′ d x = 0 {\displaystyle \int _{I}y'dx=0} and fixed volume ∫ I y d x = A {\displaystyle \int _{I}ydx=A} . The modified Lagrangian, taking into account

7722-837: The ion Ag reacts as a Lewis acid with two molecules of the Lewis base NH 3 to form the complex ion Ag(NH 3 ) 2 , which has two Ag←N coordinate covalent bonds. In metallic bonding, bonding electrons are delocalized over a lattice of atoms. By contrast, in ionic compounds, the locations of the binding electrons and their charges are static. The free movement or delocalization of bonding electrons leads to classical metallic properties such as luster (surface light reflectivity ), electrical and thermal conductivity , ductility , and high tensile strength . There are several types of weak bonds that can be formed between two or more molecules which are not covalently bound. Intermolecular forces cause molecules to attract or repel each other. Often, these forces influence physical characteristics (such as

7821-602: The liquid drop and the surface. This is sometimes referred to as the " Lotus effect ". The table describes varying contact angles and their corresponding solid/liquid and liquid/liquid interactions. For nonwater liquids, the term lyophilic is used for low contact angle conditions and lyophobic is used when higher contact angles result. Similarly, the terms omniphobic and omniphilic apply to both polar and apolar liquids. Liquids can interact with two main types of solid surfaces. Traditionally, solid surfaces have been divided into high- energy and low-energy solids. The relative energy of

7920-465: The liquid. When f = 1 and r f = r , the Cassie–Baxter equations becomes the Wenzel equation. On the other hand, when there are many different fractions of surface roughness, each fraction of the total surface area is denoted by f i {\displaystyle f_{i}} . Chemical bond A chemical bond is the association of atoms or ions to form molecules , crystals , and other structures. The bond may result from

8019-494: The micro-nano scales. In addition the sign of the line tension is not maintained through the modified Young's equation. For a sessile droplet, the free energy of the three phase system can be expressed as: At constant volume in thermodynamic equilibrium, this reduces to: Usually, the VdP term has been neglected for large droplets, however, VdP work becomes significant at small scales. The variation in pressure at constant volume at

8118-626: The molecule undergoing chemical change. In contrast, molecular orbitals are more "natural" from a quantum mechanical point of view, with orbital energies being physically significant and directly linked to experimental ionization energies from photoelectron spectroscopy . Consequently, valence bond theory and molecular orbital theory are often viewed as competing but complementary frameworks that offer different insights into chemical systems. As approaches for electronic structure theory, both MO and VB methods can give approximations to any desired level of accuracy, at least in principle. However, at lower levels,

8217-403: The nature of the chemical bond , from as early as the 12th century, supposed that certain types of chemical species were joined by a type of chemical affinity . In 1704, Sir Isaac Newton famously outlined his atomic bonding theory, in "Query 31" of his Opticks , whereby atoms attach to each other by some " force ". Specifically, after acknowledging the various popular theories in vogue at

8316-430: The net force per unit length acting along the boundary line between the three phases must be zero. The components of net force in the direction along each of the interfaces are given by: where α, β, and θ are the angles shown and γ ij is the surface energy between the two indicated phases. These relations can also be expressed by an analog to a triangle known as Neumann's triangle, shown in Figure 4. Neumann's triangle

8415-432: The past two decades (e.g. graphene , carbon nanotube , boron nitride nanomesh ). Adhesive forces between a liquid and solid cause a liquid drop to spread across the surface. Cohesive forces within the liquid cause the drop to ball up and avoid contact with the surface. The contact angle (θ), as seen in Figure 1, is the angle at which the liquid–vapor interface meets the solid–liquid interface. The contact angle

8514-434: The relative electronegativity of the atoms involved. Bonds of this type are known as polar covalent bonds . Linear function In mathematics , the term linear function refers to two distinct but related notions: In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). When

8613-462: The ring of electrons and the forces of mutual repulsion of the nuclei. The Bohr model of the chemical bond took into account the Coulomb repulsion – the electrons in the ring are at the maximum distance from each other. In 1927, the first mathematically complete quantum description of a simple chemical bond, i.e. that produced by one electron in the hydrogen molecular ion, H 2 , was derived by

8712-432: The second net force equation simplifies to the Young equation, which relates the surface tensions between the three phases: solid , liquid and gas . Subsequently, this predicts the contact angle of a liquid droplet on a solid surface from knowledge of the three surface energies involved. This equation also applies if the "gas" phase is another liquid, immiscible with the droplet of the first "liquid" phase. Consider

8811-412: The sharing of one pair of electrons. The Hydrogen (H) atom has one valence electron. Two Hydrogen atoms can then form a molecule, held together by the shared pair of electrons. Each H atom now has the noble gas electron configuration of helium (He). The pair of shared electrons forms a single covalent bond. The electron density of these two bonding electrons in the region between the two atoms increases from

8910-603: The so-called advancing contact angle, θ A {\displaystyle \theta _{\mathrm {A} }} , to the so-called receding contact angle, θ R {\displaystyle \theta _{\mathrm {R} }} . The equilibrium contact angle ( θ c {\displaystyle \theta _{\mathrm {c} }} ) can be calculated from θ A {\displaystyle \theta _{\mathrm {A} }} and θ R {\displaystyle \theta _{\mathrm {R} }} as

9009-439: The solution, as do sodium ions, as Na . In water, charged ions move apart because each of them are more strongly attracted to a number of water molecules than to each other. The attraction between ions and water molecules in such solutions is due to a type of weak dipole-dipole type chemical bond. In melted ionic compounds, the ions continue to be attracted to each other, but not in any ordered or crystalline way. Covalent bonding

9108-433: The structure and properties of matter. All bonds can be described by quantum theory , but, in practice, simplified rules and other theories allow chemists to predict the strength, directionality, and polarity of bonds. The octet rule and VSEPR theory are examples. More sophisticated theories are valence bond theory , which includes orbital hybridization and resonance , and molecular orbital theory which includes

9207-416: The surface. Contact angles greater than 90° (high contact angle) generally mean that wetting of the surface is unfavorable, so the fluid will minimize contact with the surface and form a compact liquid droplet. For water, a wettable surface may also be termed hydrophilic and a nonwettable surface hydrophobic . Superhydrophobic surfaces have contact angles greater than 150°, showing almost no contact between

9306-449: The time, of how atoms were reasoned to attach to each other, i.e. "hooked atoms", "glued together by rest", or "stuck together by conspiring motions", Newton states that he would rather infer from their cohesion, that "particles attract one another by some force , which in immediate contact is exceedingly strong, at small distances performs the chemical operations, and reaches not far from the particles with any sensible effect." In 1819, on

9405-411: The two atoms in the bond. Such bonds can be understood by classical physics . The force between the atoms depends on isotropic continuum electrostatic potentials. The magnitude of the force is in simple proportion to the product of the two ionic charges according to Coulomb's law . Covalent bonds are better understood by valence bond (VB) theory or molecular orbital (MO) theory . The properties of

9504-408: The two electrons. With up to 13 adjustable parameters they obtained a result very close to the experimental result for the dissociation energy. Later extensions have used up to 54 parameters and gave excellent agreement with experiments. This calculation convinced the scientific community that quantum theory could give agreement with experiment. However this approach has none of the physical pictures of

9603-445: The valence bond and molecular orbital theories and is difficult to extend to larger molecules. Because atoms and molecules are three-dimensional, it is difficult to use a single method to indicate orbitals and bonds. In molecular formulas the chemical bonds (binding orbitals) between atoms are indicated in different ways depending on the type of discussion. Sometimes, some details are neglected. For example, in organic chemistry one

9702-419: The velocity of a contact line is increased without bound, the contact angle increases, and as it approaches 180°, the gas phase will become entrained in a thin layer between the liquid and solid. This is a kinetic nonequilibrium effect which results from the contact line moving at such a high speed that complete wetting cannot occur. A well-known departure from ideal conditions is when the surface of interest has

9801-411: Was shown by Tadmor as, where The Young–Dupré equation ( Thomas Young 1805; Anthanase Dupré and Paul Dupré 1869) dictates that neither γ SG nor γ SL can be larger than the sum of the other two surface energies. The consequence of this restriction is the prediction of complete wetting when γ SG > γ SL + γ LG and zero wetting when γ SL > γ SG + γ LG . The lack of a solution to

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