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A Small-Angle X-ray Scattering Study of the Interactions in Concentrated Silica Colloidal Dispersions

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A Small-Angle X-ray Scattering Study of the Interactions in Concentrated Silica Colloidal Dispersions
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  A Small-Angle X-ray Scattering Study of the Interactions inConcentrated Silica Colloidal Dispersions Dong Qiu,* ,† Terence Cosgrove,* ,† Andrew M. Howe, ‡ and Ce´cile A. Dreiss †,§ School of Chemistry, Uni V  ersity of Bristol, Cantock’s Close, Bristol BS8 1TS, U.K., and Kodak European R&D, Headstone Dri V  e, Harrow, Middlesex HA1 4TY, U.K. Recei V  ed July 28, 2005. In Final Form: No V  ember 10, 2005 The structure factors of colloidal silica dispersions at rather high volume fractions (from 0.055 to 0.22) weremeasured by small-angle X-ray scattering and fitted with both the equivalent hard-sphere potential model (EHS) andtheHayter-Penfold/Yukawapotentialmodel(HPY).Bothofthesemodelsdescribedtheinteractionsinthesedispersionssuccessfully, and the results were in reasonable agreement. The strength and range of the interaction potentialsdecreased with increasing particle volume fractions, which suggests shrinkage of the electrical double layer arisingfromanincreaseinthecounterionconcentrationinthebulksolution.However,theinteractionsattheaverageinterparticleseparation increased as the volume fraction increased. The interaction ranges ( δ ) determined by the two models werevery similar. Structure factors were also used to determine the size and volume fraction of the particles. The valuesof the size obtained from the structure factors were slightly larger than those obtained from the form factors; thisdifference is ascribed to the nonspherical shape and polydispersity of the colloidal particles. The volume fractionsmeasured by these two methods were very similar and are both in good agreement with the independently measuredresults. Introduction Colloidal dispersions are widely used as model systems tostudy condensed matter physics 1 - 4 because of their size and therangeoftheinterparticlepotentials.Whenthesizeofthecolloidalparticle is very small or the density of the particle is matchedwith that of the solvent, the effect of gravity can be neglected.The phase behavior of the dispersion is then mainly determinedbytheeffectiveinterparticleinteractions.Withincreasingparticleconcentration,colloidscanformarangeofdifferent“condensed”phases including gases, liquids, and solid crystals. Whenincreasing the concentration carefully, the interparticle distancecanbecontrolled,andtheinterparticleinteractionsaremodified.At equilibrium, the interparticle interactions will determine thestructure of the condensed phase. Many studies have focused oncontrolling the ordered structure of colloidal particles bymanipulating the interparticle interactions. 5,6 Direct methods of measuring interparticle interactions incolloidalsystemsaredifficult;therefore,moreindirectapproachesneed to be used. 7 Methods such as light, neutron, and X-rayscattering have proved to be the most useful, since they allowmeasurements of both the spatial and the temporal correlationsbetween the particles. 8 Because of their short wavelength andhigh penetration into matter, neutron and X-ray scattering areespeciallyusefulinthestudyofconcentratedcolloidaldispersions.Ottewill et al. studied the electrostatic interactions in ratherdilute colloidal systems (volume fraction below 0.05) by usingsmall-angleneutronscatteringandfoundthatboththeequivalenthard-sphere model (EHS) and the Hayter-Penfold/Yukawapotential (HPY) could describe these interactions very well. 9 They have shown that for rather low volume fractions, goodagreement can be obtained between these two models, and theinteraction range wasalmostindependent of the particle volumefraction. In this work, we used small-angle X-ray scattering(SAXS)tostudytheinteractionpotentialinconcentratedcolloidalsilica dispersions (volume fraction higher than 0.05). SAXS from Colloidal Dispersions. 10,11 The scattering of X-rays from colloidal systems arises from the difference in theelectrondensitybetweentheparticlesandtheirsurroundingmedia.It has already been shown that aqueous silica colloids have agood contrast for X-ray scattering. The scattering intensity froma colloidal system,  I  ( Q ), can be written as a function of themomentum transfer,  Q , aswhere ∆ F is the contrast between the scattering units and theirsurrounding media,  V  P  is the volume of one scattering unit,  φ P is the dimensionless particle volume fraction,  P ( Q ) is the formfactor, which is determined by the size, composition, and shapeof the particle, and  S  ( Q ) is the structure factor, which accountsforthecorrelationbetweenthecolloidalparticles.Theformfactor, P ( Q ), remains constant with increasing volume fraction whenthere is no aggregation of the colloidal particles or change intheir shape (e.g., compression of an adsorbed polymeric layer),while the structure factor,  S  ( Q ), changes with the interparticle * Correspondingauthors: d.qiu@bristol.ac.uk(D.Q.);terence.cosgrove@bristol.ac.uk (T.C.). † University of Bristol. ‡ Kodak European R&D. § Current address: Pharmacy Department, King’s College London,Franklin-Wilkins Building, 150 Stamford Street, London SE1 9NH, U.K. (1) Crier, D. G.; Murray, C. A.  J. Chem. Phys.  1994 ,  100 , 9088.(2) van Blaaderen, A.; Wiltzius, P.  Science  1995 ,  270 , 1177.(3) Kegel, W. K.; van Blaaderen, A.  Science  2000 ,  287  , 290.(4) Gasser,U.;Weeks,E.R.;Schofield,A.;Pusey,P.N.;Weitz,D.A. Science 2001 ,  292 , 258.(5) Yethiraj, A.; van Blaaderen, A.  Nature  2003 ,  421 , 513.(6) Ottewill, R. H.  Langmuir   1989 ,  5 , 4.(7) Ottewill, R. H.  J. Colloid Interface Sci.  1977 ,  58  , 357.(8) Ottewill, R. H. In  Colloid Dispersions: Small-Angle Neutron Scattering ;Goodwin, J. W., Ed.; Special Publication 43; Royal Society of Chemistry:Cambridge, U.K., 1981.(9) Ottewill, R. H.; Rennie, A. R.; Johnson, G. D. W.  Ad  V  . Colloid InterfacesSci.  2003 ,  100 - 102 , 585.(10)  Neutron, X-ray and Light-Scattering: Introduction to an In V  estigati V  eToolforColloidalandPolymericSystems ,ProceedingsoftheEuropeanWorkshopon Neutron, X - Ray and Light Scattering as an Investigative Tool for Colloidaland Polymeric Systems, Bombannes, France, May 27 - June 2, 1990; Lindner,P., Zemb, Th., Eds.; Elsevier Science: Amsterdam, 1991. (North-Holland deltaseries).(11) Glatter, O.; Kratky, O.  Small-Angle X-ray Scattering ; Academic Press:London, 1982.  I  ( Q ) ) ∆ F 2 V  P φ P P ( Q ) S  ( Q ) (1) 546  Langmuir   2006,  22,  546 - 552 10.1021/la052061m CCC: $33.50 © 2006 American Chemical SocietyPublished on Web 12/15/2005  interaction and hence with volume fraction. In this study, thevolume fraction of the silica particles ranges from the dilute(0.0055) to the concentrated regime (0.165) but without theoccurrenceofaggregation;therefore,theformfactorisassumedto be constant over the volume fraction range studied. Structure Factor.  Theoretically, the structure factor can beobtainedbyFouriertransformoftheradialdistributionfunction, g ( r  ).Inturn, g ( r  )canbederivedfromstatisticalmechanicstheoryby solving the Ornstein - Zernike equation, an integral equationrelating the total correlation function and the direct correlationfunction, applying appropriate closure relations. For the hard-sphere potential, an analytical form of the structure factor hasbeen derived by Ashcroft and Lekner. 12 Hayter and Penfold 13 used the mean spherical approximation (MSA) closure to treata‘softpotential’: theHPYmodel,whichdescribesanexponentialdecay in the potential and gives the expression of the structurefactor.According to eq 1, the scattering intensity is the product of the form factor,  P ( Q ), and the structure factor,  S  ( Q ). The formfactor can be obtained experimentally by measuring a non-interacting system where the structure factor is unity. Subse-quently,thestructurefactorisobtainedexperimentallybydividingthescatteringofaconcentratedsystembytheformfactor,scaledto the same volume fractionwherethesubscripts int  and ni standforinteractingsystem(wherethe effect of interactions can be seen in the spectra) andnoninteracting system (where the effect of interactions is notshown in the spectra; normally refers to colloidal systems, inwhich either the system is extremely dilute or the interaction isscreened by some means), respectively. By fitting the structurefactors to theoretical models, the interaction potential can beobtained. Using this method, a detailed analysis of the formfactors is not necessary because they cancel out by the division.Thisapproachrequiresthattheformfactorremainconstantoverthe experimental concentration range.Equations 1 and 2 do not consider the influence of thebackground. As shown in the results section, the scatteringintensity is much stronger in the low- Q  region (0.017  ∼ 0.065Å - 1 ),andtheeffectofthebackgroundcanbeneglected;therefore,the structure factor extracted from this  Q  range is more accurateandreliable.Inthehigh Q region( Q > 0.065Å - 1 ),thebackgroundaccounts for a significant part of the scattering, and it is notproportional to the volume fraction; hence neglecting thebackgroundinequation2willnotgivethecorrectstructurefactor.For this reason, we limit our study to the low- Q  region in thespectra in this article. As it will be demonstrated, the  Q  rangeis large enough to give the first peak in the structure factors andinformation on the interparticle interaction potentials. ModelInteractionPotentials. Thehard-spherepotential(HS)isthesimplestinteractionmodelbecauseitonlytakesintoaccountthe excluded volume effect. When dealing with dispersions of charged hard-spheres, an equivalent hard-sphere model (EHS)can be devised, in which an effective hard-sphere diameter ( d  eff  )andaneffectivehard-spherevolumefraction( φ eff  )areusedinsteadoftherealdiameter( d  HS )andtherealvolumefraction( φ P - HS ). 14,15 The value of   d  eff   is larger than that of   d  HS  as a consequence of the repulsive interactions due to the existence of the electricaldouble layers. The difference ( δ HS ) between  d  eff   and  d  HS  can beregarded as the thickness of this layer as well as a measure of thestrengthandrangeofinterparticleinteractionpotential.Theseparameters are related byandIn the EHS model, the repulsive potential is considered to beinfinite when the interparticle distance is smaller than the EHSdiameter ( d  eff  ) and zero when the interparticle distance is largerthan  d  eff  . The EHS model has been used successfully to describetheinteractionsbetweensoftparticles,forexample,highlyswollenPNIPAMmicrogels 16 andblockcopolymermicelleswithahighlyswollen corona. 17,18 The HPY potential is a screened Coulombic interactionpotential that can be written aswhere  ǫ 0  is the permittivity of free space,  ǫ  is the dielectricconstant,  d  HPY  is the particle diameter for calculating the HPYpotential,  r   is the center-to-center interparticle separation, and κ  is the Debye - Hu¨ckel inverse screening length; so 1/  κ  is thedecay constant for this potential. The form of this potential isthe same as that of the Yukawa potential, which has beensuccessfullyusedtodescribebothelectrostaticandstericrepulsiveinteraction potentials. 19 - 22 It can be rewritten aswhere  U  0  is the depth of the potential. Experimental Section Materials.  Bindzil silica colloidal dispersions named as 30/360and40/220withdiametersof9and15nm,respectively,werekindlysupplied by Eka Chemicals AB (Sweden). Bindzil 30/360 has asolid content of 30 wt %, a density of 1.2 g ‚ cm - 3 , a pH of 10.0, anda Na 2 O content of 0.6 wt %. Bindzil 40/220 has a solid content of 40 wt %, a density of 1.3 g ‚ cm - 3 , a pH of 9.7, and a Na 2 O contentof 0.4 wt %, according to the manufacturer. The volume fractionmeasured for the stock silica dispersions are 0.165 for Bindzil 30/ 360 and 0.220 for Bindzil 40/220. These silica dispersions wereused without further purification. Sample Preparation.  Bindzil 30/360 was diluted with MilliQwater to give volume fractions of 0.110, 0.055, and 0.0055. Bindzil40/220wasalsodilutedwithMilliQwatertoyieldvolumefractionsof 0.165, 0.110, 0.055, and 0.0055. The most concentrated samples(0.165 for Bindzil 30/360 and 0.220 for Bindzil 40/220) were thestockdispersionsassupplied.NaClwasaddedtobothofthe0.0055volume fraction samples at a final solution concentration of 40 mMto screen the electrostatic repulsive interactions; therefore, thesesamples can be considered noninteractive, and their scattering canbe used as the form factor. (12) Ashcroft, N. E.; Lekner, J.  Phys. Re V  .  1966 ,  45 , 33.(13) Hayter, J. B.; Penfold, J.  Mol. Phys.  1981 ,  42 , 109.(14) Ottewill, R. H.; Richardson, R. A.  Colloid Polym. Sci.  1983 ,  87  , 2621.(15) Barker, J. A.; Henderson, D.  J. Chem. Phys.  1967 ,  47  , 14.(16) Stieger, M.; Pedersen, J. S.; Lindner, P.; Richtering, W.  Langmuir   2004 , 20 , 7283.(17) Pedersen, J. S.; Svaneborg, C.; Almdal, K.; Hamley, I. W.; Young, R.N.  Macromolecules  2003 ,  36  , 416.(18) Pedersen, J. S.; Gerstenberg, M. C.  Colloids Surf., A  2003 ,  213 , 175.(19) Douglas, C. B.; Kaler, E. W.  Langmuir   1994 ,  10 , 1075.(20) Zulauf, M.; Hayter, J. B.  J. Phys. Chem .  1985 ,  89 , 3411.(21) Qiu, D.; Dreiss, C. A.; Cosgrove, T.; Howe, A. M.  Langmuir   2005 ,  21 ,9964.(22) Penfold,J.;Staples,E.;Tucker,I.  J.ColloidInterfaceSci. 1997 , 185 ,424. S  ( Q ) )  I  ( Q ) int × φ ni  I  ( Q ) ni × φ int (2) φ eff  ) φ P - HS × ( d  eff  d  HS ) 3 (3) δ HS ) ( d  eff  - d  HS )/2 (4) U  ( r  ) ) π  ǫ 0 ǫ d  HPY2 ψ 02 exp[ - κ ( r  - d  HPY )]/  r   (5) U  ( r  ) ) d  HPY U  0  exp[ - κ ( r  - d  HPY )]/  r   (6)  Interactions in Silica Colloidal Dispersions Langmuir, Vol. 22, No. 2, 2006   547  Measurements.  The SAXS measurements were carried out ona NanoSTAR SAXS instrument from Bruker AXS. The instrumentusescopperK R radiation(1.54Å)producedinasealedcathodetubewith a typical current of 35 mA and a potential difference of 40 kV.The sample detector distance was set at 66.15 cm, covering a  Q range of 0.014 - 0.35 Å - 1 . The samples were placed in a sealedquartz capillary of   ∼ 1 mm path length. The sample chamber andbeampathwereevacuatedtoaround10 - 2 bar.Allthemeasurementswere carried out at ambient temperature. Transmissions weremeasured for 300 s, and the scattering was measured for a sufficientamount of time to obtain at least 1 million counts on the detector. Data Treatment.  The scattering data were converted to absoluteunits of cm - 1 using the procedure described by Dreiss et al. 23 Thestructure factors for silica samples of volume fractions higher than0.0055 were extracted from the experimental data according to eq2 by using the scattering from the 0.0055 volume fraction colloidalsilica dispersions as  I(Q) ni . As discussed above, only the low- Q  part(0.017 - 0.065 Å - 1 ) is examined in detail. The experimentallydetermined structure factors were then fitted to the theoreticalpredictions based on both the EHS potential model and the HPYpotentialmodel.Characteristicparametersdescribingtheinteraction, d  HS ,  d  eff  ,  φ P - HS ,  d  HPY ,  U  0 ,  φ P - HPY , and 1/  κ  were obtained from thefits and are discussed in the following. Results 1. SAXS Spectra  .  Scattering spectra of Bindzil 40/220 andBindzil30/360silicadispersionsatvariousvolumefractionsareshowninFigures1and2(Thespectrawerenormalizedbysilicavolume fraction to highlight the effect of the structure factors).The most dilute samples (volume fraction of 0.0055) do notshow any effect of structure factors; however, all samples atothervolumefractionshaveapeakatlow Q ,andthepeakbecomemore pronounced as the volume fraction increases. Also, theposition of peak shifts to a higher  Q  with the increase in volumefractionasaresultofthedecreaseintheinterparticleseparations.Therefore,itisreasonabletousethescatteringofthemostdilutesample(0.0055volumefraction)astheformfactortoextractthestructure factor of samples at other volume fractions by usingeq 2. At the lowest  Q  ( < 0.03 Å - 1 ), the scattering spectra showclear differences as a result of the structure factor contributionto the scattering. At higher  Q , the scattering spectra overlaybecausethestructurefactorisclosetounityandthecontributionfrom the form factor is dominant. The background from water,silica, electronic noise, and so forth is almost flat: 23 at low  Q (below 0.065 Å - 1 ), as the scattering intensity is comparablyhigh, the contribution of background is negligible; however, athigh  Q  (above 0.065 Å - 1 ), the background contributes consider-ably to the scattering intensity, and it is not proportional to thevolumefractionofsilicaparticles.Thenormalizationwithvolumefraction is not properly applicable; hence, the spectra after thenormalizationdeviate.Therefore,structurefactorsextractedfromhigh  Q  are not correct. In this article, we mainly concentrate onlow  Q  in the scattering spectra, which gives the correct profileof the structure factors. The scattering spectra of samples at avolume fraction of 0.0055 were also fitted by a spherical formfactor plus a flat background. The diameters therefore obtainedfor Bindzil 40/220 and Bindzil 30/360 from the form factors are186  (  1 and 110  (  1 Å, respectively (Table 7). 2.StructureFactors. Thestructurefactorsforthedispersionsofthetwosilicacolloidsatvariousvolumefractionsarepresentedin Figures 3 and 4. As expected, the structure factors showoscillations centered at unity, the amplitudes of which decaywith increasing  Q . It can be seen that the peaks of the structurefactors move to higher  Q  with increasing volume fraction of thesilica particles, which indicates a decrease in the interparticledistances. The amplitude of the peak in the structure factor alsoincreases with the increase in volume fraction, which reflects an (23) Dreiss, C. A.; Jack, K. S.; Parker, A. P.  J. Appl. Crystallogr. , in press. Figure 1.  SAXS spectra from Bindzil 40/220 silica dispersionsnormalized to the silica volume fraction of 0.0055. O : 0.0055; 0 :0.055;  4 : 0.110;  3 : 0.165;  ] : 0.220. The solid line is the fit of scattering from the 0.0055 volume fraction sample. Figure 2.  SAXS spectra from Bindzil 30/360 silica dispersionsnormalized to the silica volume fraction of 0.0055. O : 0.0055; 0 :0.055; 4 : 0.110; 3 : 0.165.Thesolidlineisthefitofscatteringfromthe 0.0055 volume fraction sample. Figure 3.  Experimental structure factors for Bindzil 40/220dispersions at various volume fractions.  O : 5.5%;  0 : 11.0%;  4 :16.5%;  3 : 22.0%. Dashed lines: fits to the effective hard-spheremodel.548  Langmuir, Vol. 22, No. 2, 2006 Qiu et al.  increaseintheinterparticleinteractions.Thepeaksofthestructurefactors for the larger particles (Bindzil 40/220) are located atlower  Q  than those for the smaller particles (Bindzil 30/360) atthe same volume fractions, which is expected because theinterparticle separations in the former are larger at the samevolume fraction. 3. The EHS Model.  The dashed lines presented in Figures 3and 4 are the fits to the EHS model. All the fits are in goodagreement with the data, particularly for the first peak in thestructure factor. This indicates the validity of the EHS model insimulating the electrostatic repulsions. Three independentparameters are used to fit:  d  HS ,  d  eff  , and  φ P - HS . The value of   φ eff  is calculated according to eq 3. The parameters obtained fromthe fits are presented in italicized form in Tables 1 and 2. Thevolume fraction ( φ P ) of the silica particle is also measuredindependently by evaporation, as described in the ExperimentalSection; however, to obtain the best fit, the value of   φ P - HS  wasallowed to float. For all the samples studied,  φ P  is very close to φ P - HS . Besides the interparticle interactions, the structure factoralso provides a method of measuring the particle size. All thefits were performed independently, and the sizes obtained foreachsizeofsilicacolloidaldispersionatdifferentvolumefractionsare very consistent, which emphasizes the reliability of thisapproach. In Tables 1 and 2, it can be seen that  φ eff   is alwayslarger than  φ P -  HS  , which is expected because  φ eff   includes theelectrical double layer. The contribution of the electrical doublelayers to the increase in hard-sphere radius ( δ HS ) is calculatedaccording to eq 4. For both Bindzil 40/220 and Bindzil 30/360silicacolloidaldispersions, δ HS decreaseswithanincreaseinthevolume fraction of the silica particles, which indicates that theelectrical double layer shrinks as the particles approach. This isdifferent from the findings of Ottewill et al. for the rather lowvolumefractions,inwhichtheyreportedthattheelectricaldoublelayer thickness showed little dependence on volume fraction. 9 4. The HPY Potential Model.  The dashed curves presentedin Figures 5 and 6 are the fits to the HPY potential model (eq6).Similarly,allthesefitsareingoodagreementwithexperimentaldata,especiallyforthefirstpeak.TheHPYpotentialisdescribedby four parameters: the particle size,  d  HPY , the volume fraction, φ P - HPY , the depth of the potential,  U  0 , and the decay constant,1/  κ .Asbefore,thevolumefractionusedinthefitwasalsoallowedtovarytoachievethebestfit.Theparametersusedarepresentedin Tables 3 and 4 (the parameters obtained from the fits are initalicized form). As we can see, the volume fractions obtainedfrom the fits are very close to those measured independently byevaporation. The particle sizes obtained from independent fitsarealsoveryclosetoeachother,andverysimilartothoseobtainedfrom the EHS model. The depth and the decay constant of thepotentialboth decrease with increasingvolumefractionfor bothBindzil 40/220 and Bindzil 30/360 silica dispersions. This is Figure4.  ExperimentalstructurefactorsforBindzil30/360atvariousvolume fractions.  O : 5.5%;  0 : 11.0%;  4 : 16.5%. Dashed lines:fits to the effective hard-sphere model. Table 1. Parameters Obtained from Fitting the StructureFactors to the EHS Model for Bindzil 40/220 Silica Dispersions φ P  0.055 0.110 0.165 0.220 φ P - HS  0.055 ( 0.002 0.110 ( 0.001 0.165 ( 0.002 0.220 ( 0.001 φ eff   0.131 0.214 0.254 0.297d HS  /Å  190 ( 3 194 ( 2 191 ( 2 190 ( 3 d eff   /  Å  254 ( 1 242 ( 2 224 ( 2 210 ( 2 δ HS  /Å 32 ( 2 24 ( 1 16.5 ( 1 10 ( 2   2 0.0025 0.0063 0.0032 0.0019 Table 2. Parameters Obtained from Fitting the StructureFactors to the EHS Model for Bindzil 30/360 Silica Dispersions φ P  0.055 0.110 0.165 φ P - HS  0.053 ( 0.002 0.110 ( 0.001 0.165 ( 0.001 φ eff   0.177 0.259 0.312 d  HS  /Å  140 ( 5 136  ( 2 131 ( 4d  eff   /  Å  209 ( 2 181 ( 1 162 ( 2 δ HS  /  Å 34.5 ( 3 22.5 ( 1 15.5 ( 2   2 0.0068 0.0046 0.0051 Figure5.  ExperimentalstructurefactorsforBindzil40/220atvariousvolumefractions. O : 5.5%; 0 : 11.0%; 4 : 16.5%; 3 : 22.0%.Dashedlines: fits to the HPY potential. Figure6.  ExperimentalstructurefactorsforBindzil30/360atvariousvolume fractions.  O : 5.5%;  0 : 11.0%;  4 : 16.5%. Dashed lines:fits to the HPY potential.  Interactions in Silica Colloidal Dispersions Langmuir, Vol. 22, No. 2, 2006   549  because the concentration of counterions increases with anincrease in the volume fraction of silica particles. 21 5.InteractionPotentials. Theinteractionpotentialdescribedby the EHS model are very simple and are only determined byexcluded volume effects. In this model, the interaction range isequal to the effective contribution of the electrical double layerto the hard-sphere radius,  δ HS , which is plotted against volumefractioninFigure7.AsBindzil30/360has0.6wt%Na 2 OwhileBindzil 40/220 has 0.4 wt % Na 2 O, the former has more surfacecharge as well as higher counterion concentration, and theinteraction range at low volume fraction of the former is higherbut decays faster. With the increase in the volume fraction of colloidal particles,  δ HS  decreases. As shown in the structurefactors,theorderingincreaseswiththeincreasingvolumefraction,which indicates that the interactions at the average particle - particle separation increase. In the EHS potential model, weintroduce a new parameter,  R , to estimate this interaction.  R  isdefined bywhere  h  is the average particle - particle separation (center-to-center), calculated by assuming that all the particles are placedon a cubic lattice. As  R  gives the ratio of the distance betweentwo particles to the distance at which the hard-sphere repulsiontakes place, it is expected to reflect, at least qualitatively, theinteraction at this particle - particle separation. The calculatedvalues are presented in Table 5 and plotted in Figure 8. It canbe seen that  R  increases with increasing volume fraction, whichagreeswiththeobservationsinthechangesofthestructurefactors.Therefore,  δ HS  represents the energy barrier that determines thestability of the colloidal dispersions, and  R  describes theinterparticle interaction at the average interparticle separations,which determines the structure factor.The HPY potential profiles calculated according to eq 6 arepresented in Figures 9 and 10. The interaction potential profileis a continuous curve describing the evolution of the interactionpotential as a function of the interparticle separation for a givensystem as shown in Figures 9 and 10. Using this profile, we canfind out the energy barrier of the aggregation,  U  (0), and theinteractionrange(wherethepotentialapproacheszero,normally5timesthedecayconstant,1/  κ ).Itcanbeseenthat,asthevolumefraction of the particle increases, the potential becomes weakerand decays faster (Figures 11 and 12). As  U  (0) decreases withthe increase of particle volume fraction, the colloidal stabilitydecreases. Similarly, as the surface charge and ionic strength of Bindzil30/360ishigherthanthatofBindzil40/220,theinteractionpotentialsoftheformerarehigher(higher U  (0))butdecayfaster(smaller decay constant).For an equilibrated colloidal system, the average separationbetweenparticlesisdeterminedbythevolumefraction.Therefore, Table 3. Parameters Obtained from Fitting the StructureFactors to the HPY Potential for Bindzil 40/220 SilicaDispersions φ P  0.055 0.110 0.165 0.220 φ P - HPY  0.055 ( 0.001 0.114 ( 0.001 0.165 ( 0.002 0.220 ( 0.001d  HPY  /Å  185 ( 9 196  ( 3 194 ( 3 191 ( 2U  0  /  kT 8.8  ( 1.2 6.4 ( 0.3 5.0 ( 0.2 4.1 ( 0.1 1/  κ  /Å  25.5 ( 0.4 22.9 ( 0.3 19.5 ( 0.5 16.9 ( 0.2   2 0.0081 0.0042 0.0031 0.0028 δ HPY a  /Å 32.7 ( 3.1 26.0 ( 2.4 19.9 ( 1.8 13.2 ( 2.5 a Calculated from the HPY potential by eq 8. Table 4. Parameters Obtained from Fitting the StructureFactors to the HPY Potential for Bindzil 30/360 SilicaDispersions φ P  0.055 0.110 0.165 φ P - HPY  0.054 ( 0.001 0.120 ( 0.004 0.165 ( 0.001 d  HPY  /Å 140 ( 6 137 ( 3 132 ( 2 U  0  /  kT   17.2 ( 3.1 11.2 ( 1.3 10.0 ( 1.11  /  κ  /Å 24.9 ( 0.4 18.2 ( 0.2 15.9 ( 0.3   2 0.0054 0.0029 0.0037 δ HPY a  /Å 38.0 ( 2.9 24.5 ( 2.4 19.5 ( 2.0 a Calculated from the HPY potential by eq 8. Figure 7.  δ  as a function of silica particle volume fractions givenbydifferentpotentialmodels. O : Bindzil40/220bytheEHSmodel; 0 : Bindzil 30/360 by the EHS model;  4 : Bindzil 40/220 by theHPYpotential; 3 : Bindzil30/360bytheHPYpotential.(10Åwereadded to data points for Bindzil 40/220 to separate them from thosefor Bindzil 30/360). Table 5. Interaction at Average Particle - Particle Separation Predicted by the EHS Potential Model φ P  0.055 0.110 0.165 0.220 h  /Å (Bindzil 40/220) 405 322 281 255  R  (Bindzil 40/220) 0.63 ( 0.002 0.75 ( 0.006 0.80 ( 0.007 0.82 ( 0.008 h/  Å (Bindzil 30/360) 288 228 199  R  (Bindzil 30/360) 0.73 ( 0.007 0.79 ( 0.004 0.81 ( 0.01 Figure8.  R asafunctionofthevolumefractionofthesilicaparticles. b : Bindzil 40/220;  9 : Bindzil 30/360.  R ) d  eff  h  (7) 550  Langmuir, Vol. 22, No. 2, 2006 Qiu et al.
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