The Force Between Molecules - NanoHUB

The Force between Molecules Example 2.1: 27 (Continued ) will cancel. The element of force along z, dFz , is equal to dF cos θ. The total force along z is given by integration. The charge in an area element dxdy is given by q σdx dy Qσ dx dy 1 Qq 1 2 2 4πεo (x y 2 z 2 ) 4πεo (r) z dFz dF cos θ dF 1 2 (x y 2 z 2 ) 2 a2 2b z σQ dx dy Fz (z) 3 2 b 4πεo a2 (x y 2 z 2 ) 2 2 a 2b dy σQz 2 dx 2 y 2 z 2 ) 32 b 4πεo a2 (x 2 ab σQ 1 tan πεo 2z 4z 2 b2 a2 dF The final answer is complicated and to better understand the result it is useful to make a log-log plot of Fz (z) as shown below. Two situations are plotted: (i) a small uniformly charged square plate with dimension a b 2 µm and (ii) a rectangular plate with a 4 µm, b 200 µm. The total charge on the plate in both cases is σab. Electrostatic Force (units of σQ/TTεo) 10.000 1 z 1 z2 1.000 0.100 a 4 µm b 200 µm a b 2 µm 0.010 0.001 0.01 0.1 1 10 100 1000 Distance from Plane (in µm) (Continued)

28 Fundamentals of Atomic Force Microscopy, Part I Foundations Example 2.1: (Continued ) As z increases, the force varies with z in such a way that provides insight into the interaction. When z approaches 0, the argument of the arc tangent grows without bound and tan 1 π/2. The electrostatic force is constant and equal to that expected for a point charge above a flat plane having a constant surface charge density σ, namely F Qσ/2εo . As z increases, the force decreases from this constant value. For the case of the large rectangular plate (a 4 µm, b 200 µm), the net force exhibits two well-defined regimes. The two dashed lines indicate forces that vary as 1/z and 1/z 2 . When the force varies as 1/z, the situation is approximately equivalent to the point charge Q interacting with a line charge (producing a 1/z variation). When z a, z b, the rectangular plate resembles a point charge that produces a force that varies as 1/z 2 . This example is chosen to emphasize that the variation of force with separation is often a useful way to understand and classify the type of interaction between two objects. Coulomb’s Law is often discussed in terms of an electric field that develops around a particular distribution of charge. In the simplest case for two point charges separated by a distance z in vacuum, it is very convenient generated by charge q1 at a to define the magnitude of an electric field E distance z from q1 as E 1 q1 , 4πεo z 2 F q2 E . (2.4) (2.5) The force that a charge q2 would experience if placed at a position z is then given by assuming that charge q2 does not affect charge q1 . Electric fields are vectors, just like forces, so they also have a direction associated with them. The sign convention is that electric fields point away from positive charges and they terminate on negative charges. 2.2.2 Electrostatic potential energy For two positive charges interacting with each other, an external force is required to overcome the electrostatic repulsive Coulomb force so that a

The Force between Molecules 29 If W 0, then YOU are required to do work fixed q1 F q2 z Fig. 2.4 Moving a charge q2 in the presence of a fixed stationary charge q1 requires an external force. charge q2 can be moved to a new location. The situation is indicated schematically in Fig. 2.4. In analogy with a mechanical problem requiring work to lift a mass m in a gravitational field, it is possible to define the electrostatic work required to place a charge q2 at a distance z from the stationary charge q1 . If the electrostatic force is conservative, then it is possible to define the work W (in Joules) to move an object from a reference point at infinity to some point as the line integral of the force component parallel to the displacement vector d according to z W F · d . (2.6) The work defined in this way is a signed quantity and the sign can be used to distinguish between work performed by an external agent (W 0; work done on the system) and work performed by you (W 0) in order to locate the charge at its final position. It is useful to briefly digress and consider the work performed in bringing a system to a final charge configuration (Fig. 2.4) within the context of the First Law of Thermodynamics. The First Law states dEint T dS dW, (2.7) where dEint is the change in the internal energy of the system, T is the absolute temperature of the system, dS is the change in entropy, TdS is the heat added to the system during the process under consideration, and dW is the work performed to assemble the system of charges. As seen from Eq. (2.7), the work performed can be identified with a change in internal energy only if the work is performed adiabatically (dS 0). Since a change in temperature accompanies any adiabatic process, this discussion is not particularly useful since one might expect an isothermal process (dT 0) to characterize electrostatic charging.

30 Fundamentals of Atomic Force Microscopy, Part I Foundations To remedy this situation, it is customary to define a new thermodynamic function relevant for electrostatic charging called the Helmholtz free energy F , which is defined as F Eint T S. (2.8) It follows that dF dEint T dS SdT dW SdT. (2.9) From Eq. (2.9), it is now easy to infer that the electrostatic work dW can be equated with a change in the Helmholtz free energy dF if the charging is done isothermally (dT 0). From chemical thermodynamics, the standard interpretation for a change in Helmholtz free energy is the maximum work that can be extracted from a system at some later time. In the case of electrostatic charging, we say the energy is stored in the electric field rather than in the chemical bonds. A useful quantity called the electrostatic potential energy U (z) can be defined by U (z) W. (2.10) The clear implication of Eq. (2.10) is that any work put into assembling a system of charges can be stored as electrostatic potential energy U (z), which in principle can be recovered at a later time. Calculating U (z) for relevant charge distributions will become a central focus of Chapters 3 and 4. The utility of U (z) is that it directly measures the energy required to assemble a specified charge configuration with respect to a reference specified by the lower limit of the integral that appears in Eq. (2.6). This reference is somewhat arbitrary but for the case of electrostatics, it usually corresponds to the case when all charges are infinitely far apart. Once U (z) is known, the force acting on a charge located at a position z can be calculated according to F (z) U . z (2.11) Equation (2.11) will be used extensively in the following chapters. Even though this book is concerned primarily with forces at the atomic scale, we will spend most of our time calculating energies and taking derivatives. Even though we continually speak about intermolecular forces, it is usually

The Force between Molecules 31 more convenient to calculate the interaction between molecules in terms of U (z) and then take a derivative to find the force. The results presented above for 1-dimension can be generalized to 3-dimension using vector calculus. Defining a point infinitely far away as a reference potential energy equal to 0, the electrostatic potential energy required to position a charge q at some point P can be defined as UP ( r) P r) · d . q E( (2.12) The force on the charge q at the point P is given by U ( r P ) F U (x, y, z) x x̂ P U (x, y, z) y ŷ P U (x, y, z) z ẑ, (2.13) P where x̂, ŷ, and ẑ are understood to be unit vectors pointing in the x, y, and z directions. The utility of first calculating U (r) and then taking the gradient to obtain the vector force F is an important procedural detail that should not be underestimated. For two point charges q1 , q2 separated by a distance z in vacuum, U (z) has the familiar analytical form given by U (z) 1 q1 q2 . 4πεo z (2.14) U (z) is clearly a signed quantity that depends on the polarity of the two charges q1 and q2 . Example 2.2: Plot the electrostatic potential energy of two point charges separated by a distance z. Assume a point charge Q is fixed at the origin. Consider two cases: (i) when the moveable point charge q is positive, and (ii) when the moveable point charge q is negative. A plot of the two cases is given schematically in the figure below, which shows how U (z) increases with z for the case when q is positive and how U (z) decreases with z for the case when q is negative. The starting and ending locations of the charge q are also indicated. (Continued)

32 Fundamentals of Atomic Force Microscopy, Part I Foundations Example 2.2: (Continued ) In general, if U (z) increases as a result of the motion, an external force is required to push the charge q to its final location. The final configuration is then said to be repulsive. On the other hand, if U (z) decreases as q approaches Q, the final configuration is then said to be attractive. It is important to understand this sign convention since it will be used extensively throughout our discussions. 2.3 The Forces that Hold Molecules and Solids Together Understanding the forces that hold molecules together requires a general appreciation for the origin of bonding in molecules and solids. This topic is well-established with a rich history and we make no attempt to systematically discuss it here [pauling32] [mulliken34]. For our purposes, the main results can be simplistically summarized in Fig. 2.5, which illustrates that chemical bonding varies smoothly along a continuum, rather than being sharply divided into well-defined categories such as covalent and ionic. Chemical bonding ranges from equal electron sharing between identical

The Force between Molecules 33 Fig. 2.5 The chemical bond that holds atoms together forms a continuum that spans the range shown above. On one extreme is an equal sharing of electrons between identical atoms to form covalent bonds. At the other extreme is a complete electron transfer between dissimilar atoms to form ionic bonds. atoms (H2 molecule) to the total electron transfer between atoms to form ionic compounds like NaCl. At the atomic level, whenever electrons are confined to small dimensions they can no longer be treated as a point charge. Instead they must be viewed as a delocalized wave characterized by a quantum mechanical wavefunction. The quantum wavefunction derived from Schrödinger’s equation provides the framework for describing the shape of the electronic charge cloud. Rather than reviewing the assortment of different electron wave functions, we assume the shapes of the electron orbitals are familiar from introductory chemistry courses. To describe the continuous nature of the chemical bond it becomes necessary to describe the charge distribution within a molecule. A schematic is shown in Fig. 2.6 for two cases. The redistribution of electronic charge when the two atoms are far apart is contrasted to the case when the atoms are close together at their equilibrium spacing. The electron wavefunctions are schematically indicated as uniform spherical charge clouds in this figure. When the atoms involved are identical, the final electron charge distribution must reflect this symmetry and hence the charge distribution must lie along the bisector between the two nuclei and be symmetrically distributed about it. When the two atoms involved are dissimilar, one atom will inevitably attract electrons more than the other, resulting in a final electron charge distribution that is asymmetric along the bisector between the two nuclei.

34 Fundamentals of Atomic Force Microscopy, Part I Foundations Fig. 2.6 A schematic illustration of distortions in the electron cloud during the formation of non-polar and dipolar covalent molecules. Any dipole moment p that develops provides a useful way to characterize the interaction between molecules. It is useful to identify atoms that strongly attract electrons because molecules comprised of those atoms are likely to be highly dipolar. The well-known electronegativity series for the elements provides this ranking as shown in Fig. 2.7. The chart is useful for qualitatively assessing the relative electronegativity of various elements one with respect to another. Example 2.3: Hydrocarbons are organic compounds consisting entirely of hydrogen and carbon. Would you expect hydrocarbon molecules to have a large dipole moment? By inspection of Fig. 2.7, the positions of H and C in the electronegativity chart lie roughly at the same vertical location. Therefore, neither atom is more electronegative than the other. As a result, you might expect hydrocarbon molecules to have relatively small dipole moments. Molecules with small dipole moments will not strongly interact with one another. As a consequence, when hydrocarbon molecules condense into the liquid state, you should expect the liquid to have a relatively high vapor pressure which leads to rapid evaporation.

The Force between Molecules 35 F O N C H Cl B Be P Li Mg Na Ca Sc Ti V Cr Mn Mo Fe Co Ni Cu Br S Se Si Al Zn Ga Ge I As Sn At K Rb Cs Ba Fig. 2.7 A chart of the relative electronegativity of elements in the periodic table. Elements near the top of the chart are most electronegative and have the highest affinity for attracting electrons. The most electronegative elements are F, O, N, and H, C and Cl. Source: ages/E/Electronegativity. html Last accessed December 24, 2013. 2.4 Electrostatic Forces Lead to Stable Molecules The electrostatic forces between atoms can result in the formation of stable molecules. Making precise predictions regarding the stability of a molecule requires detailed quantum mechanical calculations, which are necessary to properly describe the charge distribution accurately. It is useful to think through a simple example in order to understand the important concepts. Consider the simplest molecule H2 in which two H atoms are separated by an adjustable distance z. The H atom is represented by a positive point charge nucleus and a negatively charged quantum mechanical electron cloud, represented schematically by a spherical charge distribution as shown in Fig. 2.8. The shape of the electron charge distribution must be self-consistently calculated for every separation distance z. For the case of the H2 molecule, the charges redistribute as shown schematically in Fig. 2.6.

36 Fundamentals of Atomic Force Microscopy, Part I Foundations Fig. 2.8 A schematic illustration of the various electrostatic interactions that must be taken into account to understand how two H atoms bond to form a H2 molecule. The delocalization of the electron associated with each H atom is schematically represented by a large grey circle. For any distance z between the two hydrogen atoms, there are four interactions that must be summed. Following the polarity conventions for the electrostatic potential energy discussed in Example 2.2, the nucleusnucleus and the electron–electron interactions between H atoms are repulsive (like charges repel). On the other hand, the nucleus–electron interaction between the two H atoms is attractive (unlike charges attract). The net electrostatic potential energy will depend on the relative strength of these four contributions. An important result is that as z is decreased, the sum of the four contributions changes from negative (attractive) to positive (repulsive). Initially, for large separations, the net electrostatic potential energy is near zero. As z decreases and the two H atoms are brought closer together, the electrostatic potential energy decreases since energetically it becomes favorable for the electron charge cloud to be localized at the midpoint between the two nuclei, essentially shielding the repulsive nuclearnuclear interaction. As z continues to decrease even more, the electrostatic interaction eventually becomes positive due to the dominant repulsive interactions. An empirical model that describes this broad class of interactions between two electrically neutral atoms is known as the 12-6 Lennard–Jones interaction potential, which is often written as U (z) 4ε σ z 12 σ z 6 . (2.15)

The Force between Molecules 37 Equation (2.15) contains two adjustable parameters, which can be used to model the interaction: ε controls the depth of the minimum in U (z) and σ controls the separation at which U (z) 0. Although the 12-6 Lennard– Jones potential is widely used because of its numerical simplicity, it is known to have serious defects, especially at both small and large values of z. It is perhaps of most use to model general trends rather than specific properties of a system. A plot of the Lennard–Jones potential as a function of the nuclear separation distance z is given in Fig. 2.9 with parameters commonly used to model Ar–Ar interactions. The minimum in U (z) describes the most energetically favorable location for the two atoms and coincides with the condition of no net force. It is important to realize that different regions of the Lennard–Jones potential provide information about different processes. As an example, Region I in Fig. 2.9 is most important in high pressure situations, the shape of Region II is most important when discussing equilibrium/vibrational problems, and Region III is most appropriate for describing high temperature, thermal expansion situations. The point to remember is that different regions of the interaction potential are responsible for different physical effects. This principle will be mirrored time and again when we consider AFM tip–substrate interactions in later chapters. For the case of two Ar atoms at room temperature, thermal energies kB T ( 25 meV) exceed the depth of the potential energy well plotted in Fig. 2.9, which is about 10 meV deep, indicating that two Ar atoms will Fig. 2.9 The electrostatic potential energy U (z) (Eq. (2.15), dotted line) and the resulting force F (z) U/ z (solid line) for two Argon atoms as a function of their separation distance z. Three regimes I, II, and III are schematically indicated.

38 Fundamentals of Atomic Force Microscopy, Part I Foundations not bind to each other at these temperatures. One might expe

22 Fundamentals of Atomic Force Microscopy, Part I Foundations 2.1 Evidence for Inter-Molecular Forces An AFM senses the minute forces between a sharp tip and a nearby sub-strate. Ultimately, the net force between the tip and substrate is the sum of forces between individual atoms/molecules in the tip and atoms/molecules in the substrate.