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 ≈ ββ screening and other effects in their calculations. Thus the B12 2211 (14) Hamaker Constant A132 becomes; Thus; = (15) ∞ 12 AAA 2211 ( )− (ii ζεζε ) ( )− (ii ζεζε ) =3πh 31 32 ζ A132 d (23) For a combination of three materials when the gap between 4 ∫ ()()+ ii ζεζε ()()+ ii ζεζε 1 and 2 is filled with a medium 3, from Hamaker’s 0 31 32 calculations; Where, ε1(iζ) is the dielectric constant of material, j along −+= 2AAAA (16) the imaginary, i frequency axis (iζ) which can be obtained 131 3311 13 from the imaginary part ε ″(ω) of the dielectric constant −−+= 1 Also; 132 AAAAA 23133312 (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 132 11 33 12 AAAAA 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. involving three different materials, 1, 2 and 3, A can 132 become negative; C. Relationship between Hamaker Coefficients and p adh A132 0 (20) Free Energy of Adhesion ∆F For all given combinations, it is possible to express ∆Fadh When; f p AAandAA (21) 11 33 22 33 in terms of van der Waals energies. For instance, for a flat Or; pp (22) plate/flat plate geometry; 11 33 AAA 22 Hamaker’s approach to the interaction between adh A () −=∆ 12 (25) dF 112 2 condensed bodies from molecular properties called 12πd microscopic approach has limitations. This is true against 1 the backdrop of its neglect of the screening effect of the adh A12 molecules which are on the surface of two interacting bodies ()dF −=∆ (26) 0132 π 2 as regards the underlying molecules. Therefore, Hamaker’s 12 d 0 approach is regarded as an over simplification. For the four given combinations i.e. eqns (4) to (7) the Langbein has shown that as a consequence of the equilibrium separation distances, however, are not screening effect, in the interaction between two flat plates at necessarily the same. When a gap is a vacuum, the a separation distance d, the predominant contribution to the equilibrium separation d1 probably is but when the gap interaction by van der Waals forces comes from those parts contains a liquid, a different separation distance do may be of the interacting bodies, which are in a layer of a thickness expected. As a result of this the following becomes true; equal to the separation distance d between the two plates [6]. A +−− AAAA 132 = 33231312 (27) 2 d 0 d1 d 1 A detailed study of van der Waals forces revealed that in the case of a three-component system, the corresponding d vacuum Hamaker Constant A132 could attain a negative value given the conditions stated below. d 1 A11
Where; A11, A22 and A33 are the individual Hamaker Fig. 6. Schematic Demonstration of the Screening Effect for Constants of components 1, 2 and 3 respectively. the Interaction of Two Solid Bodies 1, at a Separation d, in The implication of this is that two adhering bodies 1 and 2 Vacuum [4] of different composition will separate spontaneously upon immersion in a liquid 3 provided the conditions given by This demonstrates the importance of surface layers like eqn (28) are fulfilled. the outer membrane of a cell, or the presence of an adsorbed layer on the overall interaction between two materials at II. MATHEMATICAL MODEL FOR THE short separations. It also demonstrates the need for the INTERACTIONS MECHANISM characterization of the surface of a body into its adhesional The mutual attraction energy, VA of two molecules in a behaviour. vacuum is given 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.
3 α 2 β ( −= )( − AAAAA ) −= −= 11 132 11 33 22 33 (36) A hvV 0 6 6 (29) 4 HH −−+= AAAAA 132 23133312 (37) Where; h = Planck’s constant A = Hamaker constant for serum (plasma) v = characteristic frequency of the molecule 33 0 A = Hamaker constant for both materials (i.e. lymphocyte α = polarizability of the molecule 13 and plasma) H = their separation distance A = Hamaker constant for both materials (i.e. the virus and The assemblies of molecules as in a solid body have 23 plasma) interaction energy as the summation of all the interaction ∞ 3 ( ) − (ii ζεζε ) ( ) − (ii ζεζε ) energies of all the molecules present and the van der Waals = πh 31 2 3 ζ A132abs d pressure, Pvdw as follows; ∫ ()()+ ζεζε ()()+ ζεζε 4 31 ii 2 3 ii A 0 (38) = 11 (30) Pvdw 3 The mean of all the values of the combined Hamaker 6πd coefficient, A132 gives an absolute value for the coefficient For a sphere of radius, R and a semi-infinite body at a denoted by A132abs; maximum separation distance, d the van der Waals force of N () attraction, Fvdw is given as; ∑ A132 RA A = 0 (39) = 11 (31) 132 abs Fvdw 2 N 6d Where A11 = Hamaker constant TABLE 1 = 2 2 βπ COMPARISON OF THE VALUES OF THE 11 qA 1 11 (32) 3 HAMAKER CONSTANTS A11, A22, A33 AND HAMAKER Where; q1 = number of atoms per cm COEFFICIENTS A132, A232 FOR THE INFECTED AND β11 = London-van der Waals constant Given two dissimilar condensed bodies of given geometry A131 FOR THE UNINFECTED BLOOD SAMPLES [8] with a separation distance, d, the corresponding van der Variable Infected Blood Uninfected Blood Waals force between them can be determined. For the system under study, the interacting bodies are the (x10-21 Peak Absolute Peak Absolute lymphocytes, 1 and the virus, 2. Joule) Value Value Value Value A11 ------1.3899 0.9659 A22 1.4049 0.9868 ------
1 2 A33 0.7052 0.2486 0.9369 0.4388 d A132 0.7601 0.2587 ------A131 ------0.9120 0.1026 A232 0.7514 0.2823 ------Fig. 7. Interaction of Two Un-identical Molecules of Lymphocyte 1 and Virus (HIV) 2, at a Separation d The table 1 shows the peak and absolute values of Hamaker constants A11 for the uninfected blood samples. A22 is the The van der Waals force between the lymphocyte, 1 and the Hamaker constant for the virus, here represented by the virus, 2 is given by the relations; infected lymphocytes. This is against the backdrop of no RA known process of isolation of the virus yet. However, it is a −= 1212 Fvdw 2 6d very close approximation for the virus owing to the manner (33) of the infection mechanism. The Hamaker constants A33 for = 2 2 βπ the plasma reveal greater values for the uninfected samples Where 11 qA 1 11 = Hamaker constant for lymphocyte which invariably indicate a higher surface energy than the infected ones. This could be validated by the surface energy = 2 2 βπ 22 qA 2 22 = Hamaker constant for the virus (HIV) approach using the contact angle method. The higher 2 = 2 qA βπ = Hamaker constant for both materials absolute values of A132 and A232 as against that of A131, as 12 12 12 well as the near zero (i.e. 0.1026x10-21Joule) value of the (i.e. lymphocyte and the virus) absolute combined Hamaker coefficient A131abs for the = βββ Where; 12 2211 uninfected samples is once again a clear indication of the Thus the Hamaker constant becomes; relevance of the concept in the HIV infection process.
= ( 2 2 )( 2 2 βπβπ ) 12 1 11 qqA 2 22 (34) III. DEDUCTIONS FOR THE ABSOLUTE COMBINED HAMAKER COEFFICIENT A = AAA 132ABS 12 2211 (35) Applying Lifshitz derivation for van der Waals forces as For a combination of the two dissimilar materials (i.e. in eqn (23 or 38); A mean of all the values of the Hamaker lymphocyte 1, and the virus 2) with the gap between them coefficients to obtain a single value known as absolute filled with plasma or serum as the medium 3 the combined combined Hamaker coefficient A132abs became necessary and Hamaker coefficient will be given by; was got as;
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.
-21 A132abs = 0.2587x10 Joule Upon deriving a mean of all values of A131 for the twenty This was done by obtaining a mean of all the values of the uninfected blood samples, an absolute value A131abs was Hamaker coefficients for the infected blood over the whole derived as given below; -21 range of wavelength, λ=230 to 890Hertz, to obtain a single A131abs = 0.1026x10 Joule value of 0.2587x10-21Joule. This value agrees with those This value is very nearly equal to zero which is a clear obtained by various authors for other biological processes as indication of the validity of the concept of Hamaker have earlier been shown in the literature review. coefficient to the process and progress of human infection with the Human Immunodeficiency Virus (HIV). The near IV. DEDUCTIONS FOR THE ABSOLUTE zero value of the A131abs shows the absence of infection in COMBINED NEGATIVE HAMAKER COEFFICIENT the blood samples thus suggesting the usefulness of the To define the condition where the absolute Hamaker concept of negative Hamaker coefficient in finding a coefficient becomes negative will require employing the solution to HIV infection. Table 1 shows the comparison of relations that express that condition. Hence, recalling eqns the Hamaker constants and coefficients for the infected and (16 to 22), we can derive a state where the Hamaker uninfected blood samples. Coefficient, A132 is less than zero. This situation could be possible with the following already stated conditions; V. CONCLUSION This research work reveals that the interactions of the A132 <0 (40) HIV and the lymphocytes could be mathematically modeled When; √A11 > √A33 and √A22 < √A33 (41) and the ensuing mathematics resolved in a bid to find a Or √A11 < √A33 < √A22 (42) solution to the infection. The values of the Hamaker coefficients and constants derived are a proof of the The mean of all values of A11 and A22 could be obtained and relevance of the concept of Hamaker coefficient to the HIV- substituted into the relation below (i.e. eqn (43)) in order to blood interactions and by extension to other biological and derive a value for A33 at which A132 is equal to zero in particulate systems. This study equally reveals the agreement with the earlier stated reasons. possibility of solving for the value of A33C (i.e. a condition ( −= )( − ) of the serum) which would favour the prevalence of a 132 11 33 22 AAAAA 33 (43) negative combined absolute Hamaker coefficient A132abs. Rearranging eqn (43) and making A33 subject of the formula Such a condition in essence would mean repulsion between 2 2 − AAA the virus and the blood cells and could prove the much = 2211 132 desired solution to the HIV jinx. A synergy of Engineers, we obtain; A33 + Pharmacists, Doctors, Pharmacologists, Medical laboratory 11 AA 22 (44) scientists etc may well be needed in interpreting the Obtaining a mean of the values of A and A to give 11 22 meaning of the A values and the medical, biological and absolute values of the Hamaker constants yielded the values 33C toxicity implications of additives in form of drugs that could given below; yield the required characteristics. A = 0.9659x10-21Joule 11 A = 0.9868x10-21Joule 22 REFERENCES Thus, plowing these values into eqn (44) and rendering A 132 [1] Omenyi, S.N., (1978): Attraction and Repulsion of ≤ 0 will give the critical value of A that satisfies the 33C Particles by Solidifying Melts, Ph.D. thesis, University condition for the combined Hamaker coefficient to be equal of Toronto, pp.23, 33, 34 to or less than zero. Hence any value of A greater than the 33 [2] Hamaker, H.C., (1937): Physica, Vol.4, p.1058 critical would be the desired value necessary to attain a [3] Van der Waals, J.D., (1873): Thesis, Leiden negative combined Hamaker coefficient. [4] Visser, J., (1981): Advances in Interface Science, Hence, the critical absolute Hamaker constant A for the 33C Elsevier Scientific Publishing Company, Amsterdam, plasma (serum) which renders the A negative is given as; 132 Vol.15, pp.157–169 A = 0.9763x10-21Joule 33C [5] London, F., (1930): Z. Physics, Vol.63, p.245 Thus for negative combined Hamaker coefficient A132 of the [6] Langbein, D., (1969): Journal of Adhesion, Vol.1, infected blood to be attained, the combined Hamaker p.237 constant of the serum (plasma) as the intervening medium [7] Dzyaloshinskii, I.E., Lifshitz, E.M., et al (1961): A33 should be of the magnitude; Advanced Physics.Vol.10, p.165 A ≥ 0.9763x10-21Joule 33 [8] Achebe, C.H., (2010): Human Immunodeficiency Inserting the above value of A33 into eqn (43) would yield a Virus (HIV)-Blood Interactions: Surface negative value for A as follows; 132 Thermodynamics Approach, Ph.D. Dissertation, A = - 0.2809x10-25Joule (when A = 0.9763x10-21Joule) 132 33 Nnamdi Azikiwe University, Awka To obtain a value for the combined Hamaker coefficient A for the uninfected blood the relation of eqns (45) and 131 (46) are employed. −+= 2AAAA 131 3311 13 (45) 2 ( −= AAA ) 131 11 33 (46)
ISBN: 978-988-19252-8-2 WCE 2013 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)