Research Journal of Recent Sciences ______ ______________________________ ______ ____ ___ ISSN 2277 - 2502 Vol. 2 ( ISC - 2012 ), 61 - 66 (201 3 ) Res. J. Recent . Sci. International Science Congress Association 61 Equilibrium Structure and Properties of Model Colloidal Suspensions Udaykumar P., Khanna T. and Behera R.N. Department of Chemistry, Birla Institute of Technology and Science, Pilani – K.K. Birla Goa Campus, Zuarinagar, Goa, INDIA Available online at: www.isca.in Received 20 th November 2012, revised 28 th December 2012, accepted 19 th January 201 3 Abstract We report the numerical results on the structure and properties of model colloidal suspensions using the hypernetted - chain (HNC) integral equation due to Allnatt, which has been successfully, applied to asymmetrical electrolyte solutions. We use the primitive model and view our system as highly asymmetrical electrolyte; the colloidal spheres are much larger and more hig hly charged than the simple ions. The variation of static correlation functions, structure factors and properties, (e.g. excess energies, osmotic coefficients etc.), is reported as a function of size, charge and concentration of colloidal particl e. The pea k position and the peak height of correlation functions show systematic trends as the asymmetry (in size, charge) increases. The effective one - component potential of the colloid (V eff ), calculated by mapping the multicomponent system to an effective one - co mponent colloidal system, is purely repulsive in line with the DLVO potential (with substantial deviations) in most of the cases. However, in some cases (with high asymmetry in charge and size, and at high colloidal concentration), V eff becomes negative. Keywords: Integral equation theory, ionic solution, Allnatt equation, hypernetted chain, effective potential. Introduction Colloidal suspensions are abundant in everyday experience and are used in countless industrial applications in the chemical, phar maceutical and food industries. The presence of solute components makes it possible to modify at will and on a large scale the static and dynamic macroscopic properties of the whole solution 1 . Different types of colloidal interactions, such as long - range r epulsive, short - range attractive, hard - sphere - like and dipolar, can be realized simply by changing the system parameters (e.g. quality of solvent, salt concentration etc.) making them excellent model systems to study a variety of phenomena (such as melting , freezing and glass transitions, etc.) in soft - condensed matter physics 2 . Colloidal suspensions of charged macroparticles dissolved in an electrolyte solution received a long im cinific in. Eximnal chniqu 3 - 6 , e.g. light, x - ray or neutron scattering view the colloidal suspensions at length scales where only the colloids (macroparticles) are visible and the rest plays the role of a b ackground. The experimental profiles have to be then interpreted by means of an effective interaction between the colloids, which indirectly includes all the effects of the remaining constituents 7 . This situation has also been reflected in the correspondin g theoretical research. Due to the large asymmetry in size, mass, and charge between the mesoscopic polyions (or macroions) and the microscopic coions and counterions, the colloidal suspension is viewed not as a mixture but as a monodisperse system formall y equivalent to a simple liquid, the colloid playing the role of atoms 8 . The key ingredient in this effective one - component picture is the determination of the effective interaction potential or force between colloids which implicitly accounts for the effe cts of the remaining constituents. One such example is the famous Derjaguin - Landau - Verwey - Overbeek (DLVO) potential 9 . Several attempts have also been made to treat macroions and microscopic ions in the primitive model on an equal footing, using the multico mponent versions of the integral equations 10 - 18 for the pair structure familiar from the theory of liquids, or resorting to computer simulations 19 . The integral equation theories have a special place in the study of macroionic fluids, because the technique s of computer simulation suffers from technical difficulties associated with very different size (and/or charge) of the solution components. Although the necessary numerical calculations are easy for simple ionic systems, they become much more difficult as the size of the hard cores and the charge asymmetry are increased 14 - 18 . Allnatt integral equation 20 is known to be very successful for dealing with ionic solutions and has been successfully used to study asymmetric macroionic solutions 21, 22 . In this repo rt, we will present the numerical calculation of correlation functions of macroionic solutions in presence of counterion using Allnatt equation. The variation of these correlations as well as some thermodynamic properties with macroion size, charge and con centration will also be discussed. Material and Methods We use the primitive model 7 to study a two component system consisting of negatively charged spherical colloid ( m ) and equivalent amount of small univalent positive ion (+) in a continuum solvent, interacting by the pair potential Research Journal of Recent Sciences ______ _ _ _______________________________ ____________ __ _ ________ ISSN 2277 - 2502 Vol. 2 ( ISC - 2012 ), 61 - 66 (201 3 ) Res. J. Recent. Sci . International Science Congress Association 62 where σ is particle diameter;  0 and  r are relative and vacuum dielectric constants, respectively. We solve the Allnatt equation 20 as given by Rasaiah and Friedman 21 along with the hypernetted chain (HNC) closure, The hat in Allnatt equation indicates the Fourier sine transformation, ρ is the number density, h ij ( r ) [= g ij ( r ) – 1] and c ij ( r ) are respectively the total and the direct correlation functions, q ij ( r ) = Z i Z j L B exp { - κ r }/ r , wh κ i h Dy screen ing length. The two component system described above can be mapped to an effective one - component macroion system (consisting of only the colloid) described by 16 with the pair potential of the effective one - component V eff (r) given b y is the so called effective direct correlation function which takes care of the effects of the small ions. Thus V eff ( r ) is that pair potential which when used in the HNC calculation in a one - component model gives the g mm ( r ) calcula ted in the multicomponent model at a given macroion radius, charge and concentration. The V eff ( r ) obtained in this report mostly obeys the form of DLVO equation, (noticeable differences at high charge, concentration and particle radius) but the DLVO parameters ( A * and κ * ) differ from that of the classical theory ( A and κ ). However, at high concentration, charge and size, the form of purely repulsive DLVO potential no longer holds: we obtained an attractive region in the function V eff ( r ). The colloidal structure factor S mm ( k ) is calculated from the Fourier transform of h mm ( r ) by S mm ( k ) = 1 + ρ m ĥ mm ( k ). The thermodynamic properties, e.g. excess internal energies, osmotic coefficients and isothermal compressibilities are obtained from the total correlation functions h ij ( r ) using standard formula 21 . Following earlier authors 21, 22 , we use the iterative method for solving the Allnatt equation, where all the functions are evaluated within a range (0, r max ). All the functions are assumed to be zero beyond this range. The range is then divided into N [= r max / ∆ r ) inval of qual iz ∆ r , and all the functions evaluated at each grid point. The periodic nature of sine and coin funcion i uilizd y imoing h condiion ∆ r . ∆ k = π/ N , wh ∆ k is the grid size in the Fourier space. We use N = 2048, ∆ r = σ mm /40. The iteration was assumed to be conversed, if h diffnc in h um of qua of τ ij ( r ) between two successive iterations becomes smaller than 1.0 10 - 8 . Results and Discussion The following abbreviations are used: R m , Z m , and C m for macroi onic (colloid) radius, charge and concentration, respectively. Similarly, g mm ( r ) (mm for macroion - macroion), g m+ ( r ) (m+ for macroion - counterion). We take R m (in nm ) = 16, 20, 25, 35, 45 and 50, Z m (in e ) = 25, 50, 75, 80, 90 and 100, C m (in μ M ) = 0.1, 0.5 , 1.0, 5.0. We use the following common parameters for all the systems: radius of monovalent ion = 0.15 nm , temperature = 298.15 K so that the Bjerrum length L B = 0.715 nm . We now describe the variation of correlation functions and properties with macroion ic charge, size and concentration. Variation with the macroion size: The radius of the colloidal particle is varied from 20 to 50 nm at fixed colloid charge of 100 e and concentration 1 M . The macroion - macroion pair correlations ( g mm ( r )) are plotted as a f unction of particle radii in figure 1. It is clear from this figure that as the macroionic radius increases, the height of the first peak of g mm ( r ) steadily increases (1.54 for R m = 20 nm to 2.36 for R m = 50 nm)) and the peak positions steadily decreases i n r (/σ  2.93 o 1.21 fo R m = 20 to 50 nm). The peaks are moving closure to the macroionic surface as we increase the size. Thus, the macroions with bigger size have lesser effective repulsion than the smaller size macroion, and thus the probability of two macroions coming closure (the peak height) increases. The increasing peaks height in the macroion - macroion structure factor ( figure 2) with increasing size of colloidal particle is also in consistent with this feature. The contact values of macroion - count erion correlation functions ( figure 1, insert) decreases as the size of macroion increases, indicating that the accumulation of counterions near the surface decreases. The accumulation of counterion at the surface of macroion depends on the surface area of the macroion and the probability of finding a counterion at the surface of macroion. It appears that with increasing radius, the second factor decreases in a greater magnitude than the increase of surface area. The effective macroion - macroion potential ( V eff ( r )), shown in figure 3 support this fact. As the size of the macroion increases, the magnitude of V eff ( r ) decreases and finally becomes negative for R m = 45 and 50 nm . Thus, the effective interaction between the macroions becomes less and less repulsi ve as the size increases and finally it becomes attractive. This type of observation in infinitely dilute macroionic solution was obtained recently 23 . Variation of macroion charge: We study the effect of increasing macroion charge at a fixed R m = 16 nm and C m = 1.0 μ M . The plots showing variations of correlation functions and structure factors are displayed in figures 4 - 6. The correlation functions obtained for Z m =25 e reproduces as reasonably well, the preliminary results of our molecular dynamic simulation 24 . The v ariation of g mm ( r ) and S mm ( k ) [ figure 4 and 5]are very much similar to that of size variation, i.e. The peak heights increases and the peak positions slightly decrease as Z m increases. However, with the increase of Z m , the magnitude of Research Journal of Recent Sciences ______ _ _ _______________________________ ____________ __ _ ________ ISSN 2277 - 2502 Vol. 2 ( ISC - 2012 ), 61 - 66 (201 3 ) Res. J. Recent. Sci . International Science Congress Association 63 g m+ ( r ) decreases [ figure 4, insert] and the magnitude of V eff ( r ) [Figure 6] first increases (from Z m = 25 e to 75 e ) and then decreases and become negative for Z m = 90 e in some portion. Thus, the effective macroion - macroion interactions increases with charge, reaches a maximu m and then become less repulsive and eventually becomes attractive (for certain inter - nuclear distances). This may be due to the fact that as Z m increases, the repulsion between macroion pair increases (which tends to increases V eff ( r ) ) and at the same ti me the condensation of counterions also increases (which tends to decrease V eff ( r )), and the later dominates at high Z m . Variation of macroion concentration: The effect of variation of macroion concentration on correlation function are studied at Z m =100e and R m =20nm. The macro ion - macro ion correlation function shown in figure 7 displays an increase in peak height and the peak height is shifted towards smaller distances as the concentration of macro ions in solution increases. This is clearly a consequenc e of increase in the repulsive interaction between the equally charged macro ions, and has been observed in several previous studies 25 - 27 . The macroion – counterion correlation function in figure 7, insert display a pronounced decrease of g m+ (r) with incre asing concentration. In Figure 8 the structure factor is displayed. It shows that with increasing concentration of colloidal particle the peak height increases and the peak position shifts towards higher inter - particle distance. It also shows secondary pea ks. The effective potential V eff is shown in figure 9. It explains that as the concentration of colloidal suspension is increased the potential decreases. The effective potential is negative for colloidal suspensions with the concentrations of 0.5M, 1M, 5M. This shows a shift towards lower inter - particle distance as the concentration increases. Variation in properties: The variation of properties excess free energy ( - Uex), osmotic coefficient (oc), isothermal compressibility (compc) and the ratios show ing deviations from DLVO potential (A*/A and κ/κ*) a a funcion of colloid iz, charge and concentration is plotted in figure 10. The excess free energy ( - Uex) decreases when size is increased while it increases when charge as well as concentration of macroion is increased. The trends in variation of osmotic coefficients (oc) are just opposite to that of – Uex. The compressibility (compc) monotonically increases as the charge of macroion increases. However, the variation is not monotonic with size and concentration. It first decreases with size and t hen starts increasing, while with increase in concentration of macroion it first increases and then decreases. The ratio ( κ/κ*) i almo on for all the cases indicating that the exponent of our V eff (r) is same as that of DLVO potential. However, we obtained substantial deviation in prefactors (A*/A deviating from one). We get maximum deviation (of A*/A) in case of size var iation and least variation in case of charge. Figure - 1 Variation of g mm ( r ) and g m+ ( r ) with the size of macroion at fixed charge ( Z m =100 e ) and fixed concentration ( C m =1.0M) Figure - 2 Variation of S mm ( k ) with the size of macroion having fixed param e ters as figure 1 Research Journal of Recent Sciences ______ _ _ _______________________________ ____________ __ _ ________ ISSN 2277 - 2502 Vol. 2 ( ISC - 2012 ), 61 - 66 (201 3 ) Res. J. Recent. Sci . International Science Congress Association 64 Figure - 3 Variation of V eff ( r ) with the size of macroion having fixed parameters as figure 1 Figure - 4 Variation of g mm ( r ) and g m+ ( r ) with the charge of macroion at fixed size ( R m =16nm) and fixed concentration at ( C m =1.0M) Figure - 5 Variation of S mm ( k ) with the charge of macroion havi ng fixed parameters as figure 4 Figure - 6 Variation of V eff ( r ) with the charge of macroion havi ng fixed parameters as figure 4 Research Journal of Recent Sciences ______ _ _ _______________________________ ____________ __ _ ________ ISSN 2277 - 2502 Vol. 2 ( ISC - 2012 ), 61 - 66 (201 3 ) Res. J. Recent. Sci . International Science Congress Association 65 Figure - 7 Variation of g mm ( r ) and g m+ ( r ) with the concentration of macroion at fixed size ( R m =16nm) and fixed charge ( Z m =100 e ) Figure - 8 Variation of S mm ( k ) with the concentration of macroion havi ng fixed parameters as figure 7 Figure - 9 Variation of V eff ( r ) with the concentration of macroion having fixed para meters as figure 7 Figure - 10 Plots of different thermodynamic properties (K/K* and A*/A (DLVO parameters), - Uex: Excess internal energy, oc: osmotic coefficient, compc: isothermal compressibility) with varying the size, charge and concentration of colloidal suspension Research Journal of Recent Sciences ______ _ _ _______________________________ ____________ __ _ ________ ISSN 2277 - 2502 Vol. 2 ( ISC - 2012 ), 61 - 66 (201 3 ) Res. J. Recent. Sci . International Science Congress Association 66 Conclusion The structure of a solution of model charged colloid is studied using Allnatt equation. 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