Diffusion of colloids at short times
a r X i v :c o n d -m a t /9707284v 1 [c o n d -m a t .s o f t ] 28 J u l 1997
Di?usion of colloids at short times
M.Watzlawek ?and G.N¨a gele
Fakult¨a t f¨u r Physik,Universit¨a t Konstanz,Postfach 5560,D–78434Konstanz,Germany
September 26,1996
[published in Prog.Coll.Polym.Sci.104,168(1997)]
Keywords:Translational Di?usion,Rota-tional Di?usion,Hydrodynamic Interaction,Charge-stabilized Colloidal Suspensions.
Abstract
We study the combined e?ects of electrostatic and
hydrodynamic interactions (HI)on the short-time dynamics of charge-stabilized colloidal spheres.For this purpose,we calculate the translational and the
rotational self-di?usion coe?cients,D t
s and D r s
,as function of volume fraction φfor various values of the e?ective particle charge Z and various concen-trations n s of added 1–1electrolyte.
Our results show that the self-di?usion coe?-cients in deionized suspensions are less a?ected by HI than in suspensions with added electrolyte.For very large n s ,we recover the well-known results for
hard spheres,i.e.a linear φ-dependence of D t
s and D r
s at small φ.In contrast,for deionized charged suspensions at small φ,we observe the interesting
non-linear scaling properties D t
s ∝1?a t φ4/3and D r s ∝1?a r φ2.The coe?cients a t and a r are found to be nearly independent of Z .The qualitative dif-ferences between the dynamics of charged and un-charged particles can be well explained in terms of an e?ective hard sphere (EHS)model.
1Introduction
Since several years,the e?ect of HI on the short-time self-di?usion coe?cients of hard sphere suspensions has been investigated in detail by various authors [1–5].For the calculation of the ?rst and second
virial coe?cients of D t s and D r
s in an expansion in terms of the volume fraction φ,both the in?uence of two-body and three-body HI was taken into account.At small φ,the currently established results for the
D t 0
=1?1.831φ+0.88φ2+O (φ3)
(1)
and by [2,5]
H r s
=
D r s
2Calculation of H t
s and H r
s
In the following,we shortly summarize the main ex-pressions needed to calculate H t s and H r s for charge-stabilized suspensions.A more detailed description of the method used by us for the calculation of short-time di?usion coe?cients is given in Refs.[7,8].
As shown in Refs.[5,9],both H t s and H r s can be measured using depolarized dynamic light scatter-ing(DDLS)from suspensions of optically anisotropic colloidal spheres.On the time scales,which are ac-cessable by DDLS,the theoretical expression for H t s is given by[3]
H t s=
1
r ,for r>
2a.Here,β=(k B T)?1,K=Z2(L B/a)(1+κa)?2, L B=βe2/?,and?denotes the dielectric constant
of the solvent.The screening parameterκis given
byκ2=L B 3|Z|φ/a3+8πn s ,where n s is the con-centration of added1–1electrolyte,and the counte-rions are assumed to be monovalent[6,7].Moreover, we use Kirkwood’s superposition approximation for g(3)(r,r′),inserting again the RMSA–g(r).Further details concerning the numerical calculation of H t s1, H r s1,H t s2,and H r s2are given in Refs.[7,8].
3Results and discussion
We focus?rst on the short-time di?usion coe?cients of deionized charged suspensions,i.e.where n s=0. Our results for H t s and H r s are shown in?gs.1and 2.The used system parameters are typical for sys-tems which have been under experimental study[11].
Obviously,the e?ect of HI on the self-di?usion co-
e?cients is less pronounced for charged suspensions
than for hard spheres at the sameφ.Furthermore, we?nd a quite di?erent volume fraction dependence of H t s and H r s for charged and uncharged particles. Whereas for hard spheres theφ-dependence of H t s and H r s is linear at smallφ(cf.eqs.(1)and(2)), we obtain from a least-square?t of our numerical results(shown as crosses in?gs.1and2)the fol-lowing results for deionized charged suspensions for
0≤φ≤0.05[8]
H t s=1?a tφ1.30,a t=2.59,(5)
0.00
0.05
0.100.15
φ
0.97
0.98
0.99
1.00H s
r
calc. points
fit: 1?1.28φ1.99
hard spheres
Figure 2:Normalized short-time rotational di?usion coe?cient H r
s for a deionized suspension with system parameters as in ?g. 1.Also displayed is the result for hard spheres according to eq.(2).
H r s
=
1?a r φ1.99,a r =1.28.
(6)
The coe?cients a t and a r are found to be nearly independent of the e?ective particle charge when Z ≥200[7,8].Note from ?g.2,that eq.(6)con-stitutes the best ?t function for H r
s (φ)even in the
extended interval 0≤φ≤0.15.In case of H t
s how-ever,the parametric form H t s =a t φp
provides no good ?t for values of φextending beyond 0.05[8].There is a simple physical explanation for the weaker in?uence of HI on the self-di?usion coe?-cients of charged suspensions as compared to un-charged ones.As already mentioned,the g (r )of deionized suspensions displayes a pronounced cor-relation hole,resulting form the strong electrostatic interparticle repulsion.Consequently,the hydrody-namic coupling between the translational or rota-tional motions of two spheres becomes rather small,thus giving rise to the observed weak in?uence of HI.Unlike charged particles,the in?uence particularly of the short-range part of HI is rather strong for hard sphere suspensions at small φ.This is due to the large probability of ?nding hard sphere particles at contact or close to the contact distance r =2a .
Along this type of arguments,it is also possible to
explain the di?erences of H t
s (φ)and H r s
(φ)in deion-ized suspensions and suspensions with nonvanishing n s .We only show here the results of our calcula-tions of H r
s (φ)for example.From these results in
?g.3,we notice that H r
s becomes more and more a?ected by HI when n s is increased.For very large
n s ,H r
s of charged particles approches the result for hard sphere suspensions,obtained semianalytically in Ref.[5](cf.eq.(2)).
This ?nding is easily explained by noticing,that the extension of the correlation hole decreases with increasing n s ,leading to a stronger hydrodynamic coupling of the particles.Upon addition of elec-trolyte,the electrostatic repulsion of the particles becomes more and more screened and short-ranged,
0.00
0.05
0.10
0.15
φ
0.90
0.92
0.94
0.96
0.98
1.00
H s
r
025*********
13000 μΜhard spheres
Figure 3:Volume fraction dependence of H r
s for various
amounts of added 1–1electrolyte,as indicated in the ?gure.All other system parameters as in ?g.1.
resulting in a pure hard-core repulsion for n s →∞[6–8].Therefore,the microstructure of the suspen-sion gradually transforms to that of hard spheres,
with H r
s approching the parametric form given in eq.(2).We further note,that the radial distri-bution function g (r )corresponding to ?g.3exhib-ites a small correlation hole even for n s =13mM .This leads to the small di?erences of our results for n s =13mM and for hard spheres in ?g.3.
We mention,that our results for H t
s show simi-lar trends,i.e.a gradual transformation of the φ-dependence of H t s from eq.(5)to eq.(1)with in-creasing n s [8].
In the remainder of this article,we focus on the
qualitatively di?erent φ-dependencies of H t s and H r
s found in case of deionized charged and uncharged suspensions.For an intuitive physical explanation,we use an e?ective hard sphere model (EHS model)[6,7],describing the actual g (r )as a unit step func-tion g EHS (r )=Θ(r ?2a EHS ).The EHS radius a EHS >a accounts in a crude fashion for the corre-lation hole,observed in the actual g (r ).We identify 2a EHS =r m ,where r m is the position of the ?rst maximum of g (r ).It is now crucial to notice that r m shows an interesting scaling property when n s =0.Due to the strong electrostatic repulsion,r m has the same φ-dependence as the average geometrical dis-tance ˉr between two spheres.Hence
a EHS ∝r m ∝ˉr =a 3
3.(7)Using the approximation g EHS (r )of g (r ),it is easy
to calculate the coe?cients H t s 1and H r
s 1in an ap-proximative way.By using far-?eld expansions of the hydrodynamic two-body mobility functions [7,8,10],one obtains the following results from the leading terms of these expansions
H t
s 1=?15a EHS
+O (a ?3EHS ),(8)H r
s 1=?5a EHS
3
+O (a ?5EHS ).(9)
This leads together with eq.(4)and(7)to the ex-pressions
H t s=1?A tφ4
3),A r>0,(11) with exponents which are in good agreement with our numerical?ndings given in?gs.1and2(cf.eqs.
(5)and(6)).
Therefore we have shown by a simple analytic calculation based on the EHS model,that the ob-served di?erences in the functional forms of H t s(φ) and H r s(φ)between charged and uncharged suspen-sions are mainly caused by the leading terms of the hydrodynamic two-body mobility functions in com-bination with the scaling property r m∝φ?1/3,valid for deionized suspensions.The higher order terms in the hydrodynamic far-?eld expansions only give rise to minor corrections to the observed scaling proper-ties depicted in eq.(5)and(6).These terms be-come increasingly important for larger volume frac-tionsφ≥0.05(cf.eqs.(10)and(11)in the EHS model).
When electrolyte is added to the suspension,eq.
(7)becomes invalid because of the enhanced screen-ing of the direct particle interactions.This causes a change in the functional behaviour of H t s(φ)and H r s(φ),as can be seen both from the EHS model and from our numerical results(cf.?g.3in case of H r s)[7,8].
Using the EHS model,it is also possible the moti-vate the nearly Z-independence of a t and a r in eqs.
(5)and(6).Since r m is nearly independent of Z for Z≥200,the EHS model predicts charge indepen-dent results for the short-time di?usion coe?cients of deionized suspensions,in agreement with our nu-merical results.
We mention,that it is also possible to deal with H t s2and H r s2within the EHS model,giving further insight in the volume fraction dependence of the dif-fusion coe?cients of deionized suspensions[7,8].It is then possible to explain qualitatively the surprising fact that H r s(φ)is well parametrized up toφ=0.15 by the functional form H r s=1?a rφ2,obtained in the EHS model by using only the leading term in the far-?eld expansion of the rotational two-body mobil-ity functions[7,8].
4Conclusion
We have presented calculations of the translational and rotational short-time self-di?usion coe?cients for charge-stabilized suspensions.The self-di?usion coe?cients of charged suspensions are less a?ected by hydrodynamic interactions than the correspond-ing coe?cients of hard spheres.As a major re-sult we have found substantially di?erent volume fraction dependencies of H t s and H r s for(deionized) charged and uncharged suspensions.The observed di?erences are well explained in terms of an e?ective hard sphere model by observing the big di?erences in the microstructure of suspensions of charged and uncharged particles.
We note?nally that recent DDLS measurements of H r s in deionized suspensions of charged?uorinated polymer particles compare favourably with our re-sults in eq.(6)[11].On the other hand,to our knowledge,no experimental data of H t s for deion-ized charge-stabilized suspensions are accessable so far.We further point out that the interesting qual-itative di?erences between charge-stabilized suspen-sions and hard spheres exist also with respect to sed-imentation[12]and long-time self-di?usion[13]. References
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