Please use this identifier to cite or link to this item: http://localhost:8081/xmlui/handle/123456789/926
Authors: Khan, Iqbal Ahmad
Issue Date: 1966
Abstract: The last part of the nineteenth and the beginning of the twentieth century witnessed many new developments in the realm of colloid science, which gradually paved the way for more comprehensive and critical studies in the years to follow. The striking observations of Schulze (l), Linder and Picton (2), Hardy (3), Zsigmcndy (4), Perrin (5), Burton (6) and others on the influence of small amounts of electro lytes on the stability of inorganic colloids and the existence of electrokinetic potentials in these colloidal systems awakened the interest of a number of eminent workers to study the problem of the stability of colloids in a quantita tive way. Not considering for the present the behaviour of the lyophilic and biocolloids, which considerably differ from the lyophobic colloids interms of electrokinetic properties (7) and stability (8) (attributed mainly to solvation), the quantitative aspect of the investigations may be divided under the following subheads: (I) Kinetics of Coagulation, (II) the nature of the electrical double layer and the sol stability interms of potential energy curves, and (III) application of conductometric and potentiometric titrations in the study of the constitution of sols. I. Kinetics of Coagulation: Inspite of the several modifications made from time to time in the Smoluchowski •s equation and the theoritical interpretations putforward to fit in the results on slow coagulation, this equation would always remain a living testimony to the marvellous imagination of Smoluchowski (9) and the great practical ingeneunlty of Zslgmondy (10). Zsigmondy (loc. cit) during the course of his studies on the action of electrolyte on gold sols distin guished two types of coagulation (as observed from the change in colour of the gold sol from red to violet), viz} slow and rapid ones. In the case of rapid coagulation the velocity is independent of the concentration, valency and nature of the coagulating ion while in the region of slow coagulation the behaviour is entirely different since the particles are not completely discharged and a residual charge always persists In particles undergoing agglomera- tion. A short resume of Smoluchowski* s fundamental studies on the velocity of rapid coagulation is worth considering. Theoritical equation for the velocity of rapid coagulation have been worked out by Smoluchowski on the basis of Zsigmondy13 assumptions. Smoluchowski considers that coagulation depends on the probability of collision between the particles and on the probability of their adhesion when collision does takes place} the second assumption is that due to collision duplets, triplets and quadriplets etc. would be formed by the collision of two singlets, one singlet and one duplet, one singlet and one triplet} the third assumption made by him is that he considers a sphere of attraction to surround each particle and postulates that collision occurs only when the centre \ of the sphere surrounding one particle enters the sphere of attraction of another particle. The minimum value for the radius of the sphere of attraction is thus twice the radius of the particles, for in such case the particles just touch when *.he centre of one enters the sphere of attraction of the other. Now as every collision between two colloidal particles leads to their adhesion, an estimate of the number of collisions in unit time gives atonce the rate of coagulation. The diffusion equation for a problem of •o. spherical symmetry is D. °(rc) • "^ (re) (i) ~dr2 -bt where D is the diffusion constant and c the concentration (number of particles in unit volume) at time t and distance r from the centre of symmetry. The solution c = cQ (1-JL), (where R and cQ are constants) of the above equation is such that at all times c • o on the surface of sphere of radius R, and c = cQ at large enough distances, r, from the centre. This solution can be obtained as shown below: The solution of the equation (i) is done by the method of separation of variables i.e. c * R(r)T(t) which means that concentration of the colloidal particle at any time is a function of radius and time t, such that R(r) is a function of r alone and T(t) is a function of t alone that is R(r) and T(t) are independent variables. ^•(rc) . 3L_ r R(r) T (t) ^t -at = r R(r) 2. T(t) "dt = r R(r) JL. T(t) dt »•» D ^(rg)^ DJL r R(r) T (t) -dr2 ih-2 [ .It- -1 =DT(t)J*.[ r R(r) 1 ~br2L J • DT(t)-dl[r R(r) 1 dr2l J = DI(t)4- X r R(r) dr [dr = DT(t)-i-T[rr..-£- R(r) + R(r) dr [ <*r ] .2 DT(t) r OPd ♦ 2 JL. R(r) dr >] dr< Substituting these values in (1) we get DT(t) r d2R(r) +2-1- R(r) dr' dr • r R(r)._ji. T(t) dt dividing both sides by rR(r).T (t) we get D r R(r) D 1— r R(r) ,2 r. QLri + 2 _sL. R(r) dr' dr - U~ . JL. ^(t) T(t) dt r.^mirl +2-1- R(r) dr2 dr ^ ; ..iii) =ife • -it T<t) •**" *"• oCis a constant • srTTx -rJ- T(t) = °C or 4 Y(tf) scCdt. Integrating it we get ivt; at TT((tt')) T(t) = •**. ec T(t) = A •' The possibilities foroCare (i)oC is real and (11)oC is \ imaginary. Now considering the case whenoCis real (positive or negative) which shows that concentration at any point should go on increasing or decreasing, which is impossible. Now considering the second case whenoCis imaginary, we get T(t) • A Sinrat + A'Cospt, this equation represents periodicity since the concentration of duplets, triplets etc. are not formed periodically but continuously. Any imaginary v&lue ofoCis, therefore, not possible. The only choice is now to putoC=:o, then we get t(t)=A. Now considering R.H.S. of equation (ii), we have —L— ail) + —2 lairl = o, multiplying by rR(r) we R(r) dr2^ rR(r) dr get, r AUI ♦ 2 dR(r) = 0 dr£ dr or -4-1* -AMxl ♦ R(r) dr dr • o or R(r) + r dR(r) _ q where G is constant dr •"• dR(r) = dr. putting G-R(r) = Z and integrating it we G-R(r) r get -j*-|Z- =fj£- - In Z • In r + In J^. putting | = E we get In Z"1 = In rE or Z= -1-, let i- =^1 rE E L :. z =IL or G-R(r) • 2L (iii) r r where G and^O are arbitary constants of a mathematical problem and have to be determined from the chemistry of the process. Taking Smoluchowski's assumptions according to which c • o at distance (r) = R and c • cQ at large enough distances. On substituting these boundary conditions we get c = cQ. (i &_) . This equation represents the r equilibrium concentration distribution around a small, perfectly absorbing sphere of radius R drawn In the bulk of colloidal solution of average concentration cQ. The 6 amount of colloidal material absorbed by the sphere in unit time is 4*R2D •(2£-) r=a • 4KRDcQ (iv) If the sphere, R, be now identified with the (average) sphere of attraction of a particle in the solution, a correction must be applied for the motion of the particles. Smoluchowski showed that the correction for the Brownlan motion of the absorbing sphere of attraction is equivalent to replacing D, in the (iv) equation by 2D. The number of collisions on a colloidal particle in unit time is thus 870RDco and the number of collisions in unit time per unit volume is 4TCRDc02. At t • o, when the coagulation may be supposed to be started, let cQ identical particles in unit volume are present. After a time t there will be concentra tions clf c2, c3 , ....of the particles formed by the adhesion of 1, 2,3f...m.... of the original particles. Considering first the single particle, they experience among themselves 4KRDc 2 collisions in unit time, and are being reduced in number at the rate 8KRDc 2 per sec. They also make 8ARD (Zc-cO c^ collisions in unit time with particles of other sizes, and thus total loss of single particles in unit time per unit volume is dci - -g^i- =8ARD(ZC-Cl) .C]L +SARDC^ (V) p The double particles are increasing at a rate 4aRDc1 in unit time and decreasing at a rate 8ARDc2"2<; per sec, (vi) :. ^1 *8KRDrC2 m— dt The triple particles are increasing at a rate 8KRDc-jC2 per sec and decreasing at a rate 8ARDcJ£c, dc i—• - dt a 8ARD ci c2 - c3^c (vii) and so on. The solution of these differential equations is cm = 2 ° , where cm represents the concentration (l+4*RDc0t)m+1 of colloidal particles after time t (following the addition of an electrolyte to the colloid), cQ represents the total number of particles initially present per unit volume. The most general form of Smoluchowski's equation cra = c° for 1+t/T rapid coagulation has been verified repeatedly by Zsigmondy (11), Westgren and Reistotter (12), Garner and Lewis (13), Muller (14) developed a mathematical application of the Smoluchowski theory to polydisperse systems in the region of rapid coagulation. Welgner and Tuorilla (15) confirmed this application.
Other Identifiers: Ph.D
Research Supervisor/ Guide: Malik, Wahid U.
metadata.dc.type: Doctoral Thesis
Appears in Collections:DOCTORAL THESES (chemistry)

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