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Mathematical Determination of the Critical Absolute Hamaker Constant

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Mathematical Determination of the Critical Absolute Hamaker Constant Proceedings of the World Congress on Engineering 2013 Vol II, WCE 2013, July 3 - 5, 2013, London, U.K. Mathematical Determination of the Critical Absolute Hamaker Constant of the Serum (As an Intervening Medium) Which Favours Repulsion in the Human Immunodeficiency Virus (HIV)- Blood Interactions Mechanism C.H. Achebe, Member, IAENG, S.N. Omenyi Abstract - HIV-blood interactions were studied using the another surface by a distance dx is given, for a reversible Hamaker coefficient approach as a thermodynamic tool in process, by; determining the interaction processes. Application was made of δw = Fdx (1) the Lifshitz derivation for van der Waals forces as an alternative to the contact angle approach. The methodology dx F involved taking blood samples from twenty HIV-infected persons and from twenty uninfected persons for absorbance measurement using Ultraviolet Visible Spectrophotometer. Fig. 1. Schematic Diagramme Showing Application of a From the absorbance data the variables (e.g. dielectric Force on a Surface constant, etc) required for computations were derived. The Hamaker constants A11, A22, A33 and the combined Hamaker However, the force F is given by; F = Lγ coefficients A132 were obtained. The value of A132abs = Where L is the width of the plate and γ is the surface free -21 0.2587x10 Joule was obtained for HIV-infected blood. A energy (interfacial free energy) significance of this result is the positive sense of the absolute Hence; δw = Lγdx combined Hamaker coefficient which implies net positive van But; dA =Ldx der Waals forces indicating an attraction between the virus Therefore; δw = γdA (2) and the lymphocyte. This in effect suggests that infection has occurred thus confirming the role of this principle in HIV- This is the work required to form a new surface of area dA. blood interactions. A near zero value for the combined For pure materials, γ is a function of T only, and the surface Hamaker coefficient for the uninfected blood samples A131abs = -21 is considered a thermodynamic system for which the 0.1026x10 Joule is an indicator that a negative Hamaker coordinates are γ, A and T. The unit of γ is Joules. In many coefficient is attainable. To propose a solution to HIV infection, processes that involve surface area changes, the concept of it became necessary to find a way to render the absolute interfacial free energy is applicable. combined Hamaker coefficient A negative. As a first step 132abs to this, a mathematical derivation for A ≥ 0.9763x10-21Joule 33 B. The Thermodynamic Approach to Particle-Particle which satisfies this condition for a negative A was 132abs Interaction obtained. To achieve the condition of the stated A above with 33 The thermodynamic free energy of adhesion of a particle possible additive(s) in form of drugs to the serum as the P on a solid S in a liquid L at a separation d [1], is given by; intervening medium will be the next step. 0 adh Index Terms - Absorbance, Dielectric Constant, Hamaker ∆Fpls (do) = γps – γpl - γsl (3) Coeficient, Human Immunodeficiency Virus, Lifshitz formula, adh Lymphocyte, van der Waal Where ∆F is the free energy of adhesion, integrated from infinity to the equilibrium separation distance do; γps is the interfacial free energy between P and S; γpl is that between P I. THEORETICAL CONSIDERATIONS and L and γsl that between S and L. A. Concept of Interfacial Free Energy For the interaction between the individual components, THE work done by a force F to move a flat plate along similar equations can be written also; Manuscript received January 29, 2013; revised February 16, 2013. adh ∆Fps (d1) = γps – γpv - γsv (4) C.H. Achebe is with the Department of Mechanical Engineering, adh ∆Fsl (d1) = γsl – γsv - γlv (5) Nnamdi Azikiwe University, PMB 5025, Awka (Phone: +2348036662053; adh e-mail: [email protected]). ∆Fpl (d1) = γpl – γpv - γlv (6) S.N. Omenyi is with the Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka. For a liquid, the force of cohesion, which is the interaction with itself is described by; ISBN: 978-988-19252-8-2 WCE 2013 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2013 Vol II, WCE 2013, July 3 - 5, 2013, London, U.K. coh ∆Fll (d1) = -2γlv (7) Hamaker made an essential step in 1937 from the mutual attraction of two molecules. He deduced that assemblies of ∆Fadh can be determined by several approaches, apart from molecules as in a solid body must attract other assemblies. the above surface free energy approach. The classical work The interaction energy can be obtained by the summation of of Hamaker is very appropriate [2]. all the interaction energies of all molecules present as in To throw more light on the concept of Hamaker fig.4 below. Constants, use is made of the van der Waals explanation of 4 . the derivations of the ideal gas law; . 3 PV = RT (8) 2 1 1 It was discovered that the kinetic energy of the molecules d which strike the container wall is less than that of the bulk molecules. This effect was explained by the fact that the 1 surface molecules are attracted by the bulk molecules (as in 1 fig.2) even when the molecules have no permanent dipoles. 2 It then follows that molecules can attract each other by some . kind of cohesive force [3]. These forces have come to be 3 . known as van der Waals forces. van der Waals introduced the following corrections to eqn (8); Fig. 4. Interaction of Two Semi-infinite Solid Bodies 1 at a a Separation d in Vacuum [4] + () =− (9) P 2 RTbV V This results in a van der Waals pressure, Pvdw of attraction a between two semi infinite (solid) bodies at a separation The correction term to the pressure, indicates that V 2 distance, d in vacuum; A the kinetic energy of the molecules which strike the = 11 Pvdw 3 (11) container wall is less than that of the bulk molecules. This 6πd signifies the earlier mentioned attraction between the surface For a sphere of radius, R and a semi-infinite body at a molecules and the bulk molecules. minimum separation distance, d the van der Waals force, Fvdw of attraction is given by; RA −= 11 Fvdw 2 (12) 6d Where A11 is the Hamaker Constant (which is the non- geometrical contribution to the force of attraction, based on v molecular properties only) Fig. 2. Attraction of Surface Molecules by Bulk Molecules in a Container of Volume V [4] R After the development of the theory of quantum mechanics, London quantified the van der Waals statement vacuum d for molecules without a dipole and so molecular attraction forces began to be known as London/van der Waals forces 1 [5]. London stated that the mutual attraction energy, VA of two molecules in a vacuum can be given by the equation; α 2 β −= 3 υ −= 11 Fig. 5. Interaction of a Sphere of Radius R at a Separation d A hV 0 6 6 (10) 4 HH from a Solid Surface of the same Material 1 in Vacuum [4] The interaction of two identical molecules of a material 1 is According to Hamaker, the constant A equals; shown in fig.3 below. 11 = 2 2 βπ 11 qA 1 11 (13) α 1 α 1 3 H Where q1 is the number of atoms per cm and β11 is the London/van der Waals constant for interaction between two molecules. Values for β can be obtained in approximation Fig. 3. Interaction of Two Identical Molecules of Materials from the ionization potential of the molecules of interest, 1 and Polarizability α, at a Separation H [4] and so the Hamaker Constant can be calculated. The corresponding van der Waals force between two condensed ISBN: 978-988-19252-8-2 WCE 2013 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2013 Vol II, WCE 2013, July 3 - 5, 2013, London, U.K. bodies of given geometry can be calculated provided their The limitations of Hamaker’s approach led Lifshitz et al separation distance is known. to develop an alternative derivation of van der Waals forces For combination of two different materials 1 and 2 in between solid bodies [7]. The interaction between solids on approximation: the basis of their macroscopic properties considers the ≈ β β screening and other effects in their calculations. Thus the B12 11 22 (14) Hamaker Constant A132 becomes; Thus; = (15) ∞ AAA12 11 22 ε(i ζ)− ε( i ζ)ε(i ζ)− ε( i ζ) =3πh 1 3 2 3 ζ A132 d (23) For a combination of three materials when the gap between 4 ∫ ε()()i ζ+ ε i ζ ε()()i ζ+ ε i ζ 1 and 2 is filled with a medium 3, from Hamaker’s 0 1 3 2 3 calculations; Where, ε1(iζ) is the dielectric constant of material, j along AAAA= + − 2 (16) the imaginary, i frequency axis (iζ) which can be obtained 131 11 33 13 from the imaginary part ε ″(ω) of the dielectric constant = + − − 1 Also; AAAAA132 12 33 13 23 (17) ε1(ω). Rewriting these equations will give; The value of A11 could be obtained from the relation; 2 2 ε − 2 2 − AAA=( − ) (18) 10 1 n1 1 131 11 33 A = 5.2 = 5.2 (24) 11 ε + 2 − And; 10 1 n1 1 =( − )( − ) Where ε is the dielectric constant and n the refractive AAAAA132 11 33 12 33 (19) 10 1 index of the polymer at zero frequency, both being bulk Equation (19) shows that, for a three-component system material properties which can easily be obtained.
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