What molecular weight polymer is necessary to provide steric stabilization? = [1]

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1 1/7 What molecula weight polyme is necessay to povide steic stabilization? The fist step is to estimate the thickness of adsobed polyme laye necessay fo steic stabilization. An appoximation is: 1 t A d 48kT 11 = [1] whee t is the necessay polyme thickness, A 11 is the Hamake constant fo paticles of type 1 in a solvent of type, d is the paticle diamete, and kt is the Boltzmann facto. The next question is: What molecula weight of polyme will give that adsobed laye thickness? A easonable fist appoximation is that polymes which ae adsobed and also act as steic baies ae those that ae soluble, but baely so. This means that the polyme is in its theta condition, both in solvent composition and tempeatue. Fotunately, the size of polymes in thei theta conditions is well studied. In theta conditions the polyme inteacts with itself about the same as it does with the solvent so that it takes on a andom configuation (i.e., the second viial coefficient is zeo). Fo many common linea polymes the mean end-to-end distance is given by the expession: 1/ 1/.6 MW [] Whee 1/ is in nanometes and MW is the polyme molecula weight. Table 1 gives examples. 1 Moison, I.D.; Ross, S. Colloidal dispesions: Suspensions, emulsions, and foams; John Wiley & Sons: New Yok;, pp Nappe, D.H. Polymeic stabilization of colloidal dispeions; Academic Pess: New Yok; 1983; pp Ian Moison 5

2 /7 Table 1. Typical dimensions of polyme chains of diffeent molecula weights (andom coil appoximation) Molecula weight End-to-end distance (nm) 1/ 1, 1, 6 1, 1,, 6 Numeous theoies and Monte Calo calculations have aived at the conclusion that, to a easonable appoximation, the mean end-to-end distance fom the suface is twice the mean end-to-end length in solution. This makes some sense since the andom configuations of a polyme fom a suface ae hindeed by the pesence of the suface and so each futhe. Theefoe the thickness of the adsobed laye, t, is about twice the mean end-to-end distance given in Table 1. An useful equation is gotten by combining Equation [] with [1]: MW A11 = d.9kt [3] The necessay molecula weight scales as the squae of the paticle size. In a polydispese dispesion, only the small paticles might be stabilized if the polyme molecula weight is low. Sophisticated theoies ae available to estimate the size of a polyme, but fo pactical applications a poductive pocedue is to measue the size of the polyme in solution. Once the size of an inteesting polyme is known, especially as a function of solvent composition, then moe compehensive ideas can be intoduced like the numbe and stength of attachment points and the flexibility of the polyme chains. Ian Moison 5

3 3/7 1. Measuing the size of a polyme in solution The size of a polyme in solution is measued eithe (o both!) by heology and by light scatteing. 1.1 RHEOLOGY Two simple heological methods ae used to measue the size of a polyme in solution: 1.1.a Fom the viscosity of dilute polyme solutions In 196 Einstein published an impotant elation descibing the viscosity of a dilute suspension of paticles 5 η = η 1 + Φ+ O ( Φ ) +... [4] whee η is the viscosity of the suspension, η is the viscosity of the solvent, and Φ is the volume faction of paticles. The equation has been modified fo all kinds of dispesed phases, e.g. dops, bubbles, and polymes. 3 But we can make good use of this fom of the equation. Re-aanging Equation [4] gives: η η Φ lim [5] 5 c η The elation between volume faction and mass concentation is: 3/ N Φ c [6] MW 3 Goodwin, J.W.; Hughes, R.W. Rheology fo chemists: An intoduction; Royal Society of Chemisty: Cambidge;, pp Ian Moison 5

4 4/7 Whee N is Avogado s numbe and c is the mass concentation. Substituting Equation [6] into [5] and e-aanging gives: MW 1 η η lim [7] 5N c η 3/ c o 3/ 5 MW N [ η] [8] whee [η] is the intinsic viscosity defined by: 1 η η = c c η [ η] lim [9] The intinsic viscosity of a polyme is detemined by measuing a seies of polyme solutions ae gadually geate and geate dilution. This is often done with commecial equipment b Fom the viscosity at polyme concentations Intinsic viscosity is the measue of the size of polymes at concentations so low that thee ae no polyme-polyme inteactions. At highe concentations, inte-molecula inteactions become impotant and the viscosity inceases faste. At the concentation whee the viscosity inceases faste than the Einstein limit, the polyme chains ae close enough togethe that they stat to inteact, a concentation commonly efeed to as c *. A easonable assetion is that the polyme chains inteact when they fill-space. Fo ou puposes this means that c* and the intinsic viscosity ae simply elated: 4 I have used equipment fom Viscotek ( with good esults. Ian Moison 5

5 5/7 1 c* = [1] [ η] c* is easy to measue. The viscosity is measued as the polyme concentation is inceased. A log-log plot of the data is made and the concentation when the slope of the data becomes geate than one is taken as c*. Combining Equation [1] and Equation [8] gives a second appoximation fo the polyme size in solution: 3/ MW 5 Nc* [11] Theefoe thee ae two methods to estimate the polyme size in solution by viscosity: the fist with solutions appoaching infinite dilution using intinsic viscosity and the second using c*. 1.1.c Fom simple polyme theoy: The intinsic viscosity of a polyme solution is a well-studied physical popety. A commonly used elation fo polymes is the Floy elation: [ η ] = a KMW i [1] whee K and a ae polyme popety fom polyme handbooks. 5 MW is the mass aveage molecula weight. This value of intinsic viscosity can be used in Equation [8]. Othe methods to get the "size" of the polyme ae found in Tables in the Polyme Handbook. Chapte VII in the 3 d edition gives Table D. Calculated unpetubed dimensions of feely otating chains and Table E. Unpetubed dimensions of linea polyme molecules, whose values ae explained in the peceding section in that Chapte. 5 Polyme Handbook, 4 th edition edited by J. Bandup, E.H. Immegut, E.A. Gulke, A. Abe, and D.R. Block; John Wiley & Sons, New Yok; 3, in two volumes. In the 3 d Edition the Chapte is: Viscositymolecula weight elationships and unpetubed dimensions of linea chain molecules by Kuta, M.; Tsunashima, Y. Ian Moison 5

6 6/7 1. LIGHT SCATTERING A seies of dilute polyme solutions ae pepaed. The change in efactive index with polyme concentation and the intensity of light scatteed as a function of angle ae measued. The data is analyzed by means of a Zimm plot and the molecula weight and adius of gyation obtained. 6 Fist the concentation dependence of the efactive index at constant tempeatue and pessue, ( δ n ), is measued and a paamete, K, is calculated by: δ c Tp, K 4π n δ n = λ δc 4 N Tp, [13] whee λ is the light wavelength in vacuo, N is Avogado's numbe, and n is the efactive index of the solvent. The intensity of light scatteed, called the Rayleigh atio, is measued: θ = [14] R(, c) I I whee θ is the scatteing angle, is the distance fom the sample to detecto, I is the intensity of the scatteed light, and I is the intensity of the incident light. The data eduction is by a method intoduced by Zimm 7. The elevant equations solved ae: 6 Moison, I.D.; Ross, S., p 6. 7 Keke, M. The scatteing of light and othe electomagnetic adiation; Academic Pess: New Yok; 1969; pp Ian Moison 5

7 7/7 θ 16 sin Kc 1 π g = 1 + as c R( θ, c) MW 3λ [15] and 1 Kc = lim [16] MW R c c θ ( θ, ) which give both the molecula weight, MW, and the squaed adius of gyation, the polyme in that solvent at that tempeatue. g, of All this can be automated. 8 8 A ecommended piece of equipment to make the measuements and calculations automatically is available fom Wyatt Technologies ( Ian Moison 5