An experimental investigation of linear rheological properties of monodisperse ordered latex dispersions has been carried out. Measurements of the dynamic shear modulus and the dynamic viscosity in the frequency range between 0.04 Hz and 2500 Hz are reported. The volume fraction and the electrolyte concentration were variables in the experiment. The results show that a description of the rheological behaviour with a simple linear phenomenological equation is not possible. The measured dynamic moduli can only be described with a great number of relaxation times

Blom, C.

Lopulissa, J.S.

Mellema, J.

Reuvers, A.J.

English

NARCIS 

Frequency dependent linear viscoelastic properties of ordered polystyrene latices

University of Twente Research Information

Colloid & Polymer Science Colloid & Polymer 8r 262:397-402 (1984) COLLOID SCIENCE Frequency dependent linear viscoelastic properties of ordered polystyrene latices C. Blom, J. Mellema, J. S. Lopulissa and A.J. Reuvers Rheology Group, Department of Applied Physics, Twente University of Technology, Enschede, The Netherlands Abstract: An experimental investigation of linear rheological properties of monodisperse ordered latex dispersions has been carried out. Measurements of the dynamic shear mo- dulus and the dynamic viscosity in the frequency range between 0.04 Hz and 2500 Hz are reported. The volume fraction and the electrolyte concentration were variables in the ex- periment. The results show that a description of the rheological behaviour with a simple linear phenomenological equation is not possible. The measured dynamic moduli can only be described with a great number of relaxation times. Key words: ordered polystreen latex, viscoelastic properties 1. Introduction A monodisperse latex dispersion is a frequently used model system for the investigation of electrostatic interacting forces between colloidal particles. The present study is confined to linear viscoelastic proper- ties of ordered polystyrene latices. As such it is con- nected to other investigations on the rheological pro- perties of polystyrene latex dispersions reported in the past years [1-10]. In these studies on relates the viscoe- lastic properties of the latex dispersions to the interpar- tide forces, which are dependent on parameters such as particle diameter, electrolyte concentration and sur- face potential. The viscoelastic properties derived in these studies are only valid in a very small frequency range. One group has measured the dynamic shear modulus at about 5 Hz. [9]. A few groups have mea- sured the complex dynamic modulus at one frequency between 185 and 350 Hz [1, 6, 7], another group at 40 kHz [4] and 70 kHz [2]. Basic for their analysis is the question if the measured shear modulus can be compared with the theoretical shear modulus they de- rived. Russel and Benzing [5, 6] state that their theory (based on a cell model) and measurements at about 300 Hz give the low frequency plateau value of the dy- namic shear modulus. Buscall et al. [7, 8] indicate the shear modulus calculated by considering pair wise added particle interactions as the high frequency limit. They conclude to have measured it between 185 and 225 Hz for r > 0.14. Mitaku et al. [3, 4] also used a cell model but reason to have measured the high frequency limit at 40 kHz ("glass-like state"). In the present study the dynamic moduli of monodisperse polystrene la- tices are measured between 0.04 Hz to 2500 Hz. Its aim is to show that the knowledge of this dynamic be- haviour in a large frequency range can give more in- sight in the number and nature of the relaxation me- chanisms that play a role in these systems with stron- gly interacting particles. 2. Experimental 2.1 Latex preparation The polystyrene latex was prepared by polymerization of sty- rene in aqueous potassium hydrogen carbonate (KHCO3) at 70 ~ using potassium persulphate (K2 $208) as initiator. The used me- thod has been described in detail by Norde [11]. The water was deionised, bidistilled and saturated with N2. The styrene (Baker grade) was distilled under reduced N2 pressure. The potassium per- sulphate (Merck p. a.) was recrystallized, washed with cold water and dried. The potassium hydrogen carbonate (Merck p. a.) was used without further purification. The concentrations of the com- ponents were KHCO3 : 10 -3 mol/dm ~, K2S208 : 18.510-3 mol/dm 3 and 65.6 cm 3 styrene on a total of 4 dm a mixture. The polymeriza- tion time was 24 hrs. The concentration and cleaning of the latex was performed with a diafihration system (Amicon TCF 10). The diafiltration of the latex under N2 atmosphere was repeated until the conductivity of the filtrate was less than 2/lmho. The conduc- tivity of the used wash water, with quality as given above, was less than 1/lmho. The volume fraction of the latex after purification was about 0.25. The latex volume fractions were determined gravimetrically, by drying a sample to constant weight at 60 ~ The volume fraction W 726 398 Colloid and Polymer Science, VoL 262" No. 5 (1984) was calculated by taking the density of polystryrene to be 1.05 103 kg/m 3. Three sample with volume fractions of 0.25, 0.21 and 0.10 were prepared from the same batch of material. The mean diameter of the latex particles, as determined with electron microscopy was 330 nm, with a standard deviation of 3 %. 2.2 Rheological measurements The dynamic moduli were determined with four single torsion pendula operating in the frequency range of 80 Hz to 2500 Hz and with a dynamic viscometer operating in the frequency range of 0.04 Hz to 1 Hz. A detailed description of the torsion pendula is gi- ven by Blom and Mellema [12]. The measuring method is based on the determination of the resonance frequencies and bandwidths of the pendula immersed in the liquid under investigation and in air. From the frequency and bandwidth difference between the measu- rements in the two media, the dynamic moduli can be deduced. A condition to be met for this method is that the amplitude of the induced shear wave is almost damped out when it reaches the wall of the sample holder. The penetration distance i. e. the distance over which the amplitude is damped out by a factor or 1/e is given by [13]: 1 , / 2  [ G '2 + G "2 \ O=~V~t  CG7~ +G,,2_G,-) , (1) where 0 is the sample density and ca is the angular frequency. The form of this formula differs appreciably from the one given by Benzing and Russel [6], but they are identical apart from a typo- graphical error. The dynamic viscometer used in an apparatus simular to the one described by Te Nijenhuis [14]. This apparatus consists of a cylin- der suspended by a torsion wire. The torsion wire is given a sinusoi- dal oscillation. When the cylinder is immersed in the sample it will oscillate with the same frequency but with a different amplitude and out of phase. From the phase lag and the amplitude ratio of the top of the torsion wire and the cylinder, the dynamic moduli can be cal- culated. For this method a condition to be met is that the wave- length of the induced shear wave is much larger than the gap bet- ween the measuring cylinder and the surrounding cylindrical sample holder. The expression for the wavelength;t is given by [13]: co ~G '2 + G "2 + G'- " (2) The steady state viscosities were determined with a Couette vis- cometer (Contraves LS 30). All measurements were performed at a temprature of 10.0 _+ 0.5 ~ Except for the steady state measurements, the measurements were performed under a Nz atmosphere. When an ordered latex is exposed to air, the ordering will disappear within a few days (see al- so [15]). Then the latex changes from bright iridescence to milky white. 3. Results 3.1 Steady state measurements Due to the fact that the steady state measurements could not be performed under a N 2 atmosphere, these measurements gave no reproducible results for the steady state viscosity. The magnitude of the measured viscosities is of the order of the measured dynamic vi- cosities at the lowest frequency. All the steady state measurements showed the same phenomenon: with increasing strain a sudden drastic lowering of the vis- cosity occurred at a strain of about 0.25. 3.2 Frequency dependence of G* A. Influence of the volume fraction on G* Of the prepared samples, with volume fraction 9 = 0.25, 9 = 0.21 and 9 = 0.10 the dynamic modulus G *  = G' + iG"  =- O(rl" + irl') =- icorl*, (3) was measured. In figure 1 the dynamic shear modulus G' is given as function of the frequency. In figure 2 the relative dynamic viscosity, defined by /~1re 1 =/77// (4) is given. Here r/ is the viscocity of the continuous phase. In these and the following figures the lines are drawn tentatively. Both conditions mentioned in sec- tion 2 for the damping and the wavelength are met for the results given here. 10 0.1 / Io -2 io 4 io I io3 Ir FREQUENCy (HZ) Fig. 1. The dynamic shear modulus as function of the frequency; the indicated parameter is the volume fraction 9 of the latex Blom et aL, Viscoelastic properties of polystyrene letex 399 ~'REL 103 lO2 101 --0.i 10o i 10-2 10-1 10 0 101 10 2 10 3 10 4 FREQUENCY (Hz) Fig. 2. The relative dynamic viscosity as function of the frequency; the indicated parameter is the volume fraction (it) of the latex. The calculated high frequency limits are indicated with bars number will become smaller which means that ma- croscopically the latex behaves less and less as a conti- nuum. The measurements give the impression that the ratio,i/a is large enough, as no anomalies at the highest frequencies are abserved. Figures 1 and 2 show, as ob- served by others [4, 6, 7], that at a fixed frequency (7' and r/increase as function of the volume fraction. New is that it appears that (7' is increasing with frequ- ency. Further r/re1 is strongly decreasing with frequ- ency. Noteworthy is that c9 log (7' is almost independent of a logF the latex volume fraction. B. Influence of  the electrolyte concentration on G* To the sample with volume fraction (it) = 0.21, KC1 has been added such that the overall KC1 concentra- tions in the sample were about 10 -5,10 -4 and 10 -3 mol/ dm 3 respectively. In figures 3 and 4 the influence of the added electrolyte on the dynamic shear modulus and the relative dynamic vicosity as function of the frequ- ency is demonstrated. The upper curve in both figures represents the measurements with no electrolyte add- ed to the sample. The gradual destruction of the lattice with increasing electrolyte concentration displays it- The dynamic viscometer measurements were per- formed with a maximum strain Ymax < 0.1 and for the 100 torsion pendula measurements Ymax < 0.01. Linearity was checked by changingymax by a factor 2 and 10 re- spectively below these upper limits. No deviation from linearity has been demonstrated. 10 The estimated uncertainties in the dynamic measu- rements are summarized below. For the dynamic vis- cometer the uncertainty in 1"/' is 10 0/0 with a bound of lO-3PasandtheuncertaintyinG'is lO~ 1 of 2.10- 2 N / m  2. For the torsion pendula the uncertain- ty in 17' is 5 % with a bound of 5.10 -5 Pa s and the un- certainty in O' is 5 % with a bound of 5.10 -2 Nm 2. The maximum feasible frequency is related to the ratio of the wavelength,~ of the shear waves and the in- 0.1 terparticle distance a. To provide that macroscopically the latex can be considered as homogeneous in good approximation, this ratio ,~/a should be much larger than unit. For example for 9 = 0.1 (for this value the interparticle distance is about equal to the particle dia- meter) and frequency F = 2500 Hz, The values for G' and G" are about 3,5 N / m  2 and 25 N / m  2 respectively. From (2), using Q=103kg/m 3 it follows that ;t 10-4 m, which means that there are about 150 particles over one wavelength. For higher frequencies this 0.01 G' (N/M 2) 10 -1 10 o 101 102 10 3 FREQUENCY (Hz) Fig. 3. Influence of the added electrolyte on the dynamic shear mo dulus, (it) = 0.21; indicated parameter is the (added) electrolyte con- centration in mol/dm 3. 400 Colloid and Polymer Science, 1/ol. 262. No. 5 (1984) lo 3 lo2 I~' REL lO 1 loo 10-2 "10-1 100 101 102 1~ 10 q FREOUENCY (HZ) Fig. 4. Influence of the added electrolyte on  the relative dynamic  viscosity; q~ = 0.21; indicated parameter  is the (added) electrolyte concentrat ion in m o l / d m  ~ self in the decreasing values of G' and/'])el at  fixed fre- qencies. It is interesting to observe that 0 log G' and 01ogF 0log ?]rel change appreciably by addition of electro- ,01ogF lyte. 4. Discussion 4.1. Linearity The meausrements of the steady state viscosity made clear that the strain range in which linear beha- viour is observed is small (y < 0.25). Obviously this is related to the interparticle distance in the lattice and one should expect deviations from linear behaviour above an upper limit ofy which will be less than 0.5. The dynamic measurements have been performed such that 7max < 0.1 for all frequencies and below this upper limit no deviation from linearity was demons- trated. 4.2. Pheonomenological equation Given the linear viscoelastic behaviour, the scope of the analysis is restricted to the general linear viscoelast- ic constitutive equation (see e. g. [16]): ~ - ~  d k ~ - ~  d k . 2. qh~e (5) ~ pk ~iFO = + qe . k=O k=O de Here o is the shear stress, e is the strain, ~ = ~ - ,  Pk, qk, and q are parameter, and t is the time. Four catego- ries of materials can be indentified. Two of them are called solids because q 4= 0. For the other two q = 0 and these are therefore called fluids. For both categories, solids and fluids, two relationships are allowed bet- ween m and n (rn = n + 1 and rn = n). For harmoni- cally oscillating shear experiments with angular frequ- ency co, the stress and strain can be written as o = 0 0 e ic~ + r and e = e0 e loot, where r is a phase shift. In this case relation (5) transforms into 2 Pk (ico) k o = 2 qk (ico) ke + qe. (6) k = 0  k=O The dynamic modulus defined by O G* ~ -  (7) C is given by n q + ~ qk (ico) ~ +1 k=0 (8) G*(ico)- 2 Pk(ico) h k=O Relevant to the analysis is the behaviour of G* (c. q. q*) for c0--, 0 and for co---, oo. In Table 1 a survey is given of the presence (indicated by a plus sign) or the absence (indicated by a naught) of some quantities. Considering the behaviour for co--, 0 one realizes that the lattice of the latex particles is temporary, i. e. due to the thermal motion there is a finite chance that a Tabie 1. Survey of some  propert ies of  the four  different types  of  li- near viscoelastic materials q r/' G'  r/' G '  fluid m = n + 1 0 + 0 0 + fluid m = n 0 + 0 + + solid m = n + 1 + + + 0 + solid m = n + + + + + Biota et aL, Viscoelastic properties of polystyrene latex 401 particle leaves its place, so that lim G~,_0=0. For co--, oo there is always dissipation in the systeme, e. g. by the liquid between the particles, so that lim r/~_ co 4= 0. It can be concluded (see table 1) that the va- lid constitutive equations is given by / /  p k - ~  o = q k ~ i ~  ~ (9) k=0 This conclusion differs from that of Buscall et al. [7] who deduced from their measurements a fluid type behaviour with rn = n + 1 and one retardation time and two relaxation times. Since their model is not compatible with lim r/~,~ = 4= (3 we think that even the fluid type with m = n + 1 is not relevant for (ordered) latices. In addition our measurements show that a great number of relaxation times is necessary to des- cirbe the viscoelastic behaviour. 4.3. Possible relaxation mechanisms At low electrolyte concentrations, the latex par- ticles order in a lattice. Assuming the sample not to be one single crystal, ordering exists over some characte- ristic length which probably decreases with increasing electrolyte concentration. There is no reason to expect the ordering to be perfect, so that dislocations, vacan- cies and interstitials will be present. Calculation of a shear modulus by means of pair wise added neighbour interactions for these different kind of configurations would give a range of G'-values with pertinent relaxa- tion times and strengths. This would partly explain the presence of many relaxation times. The actual situa- tion, however, incorporates mechanisms which can not be accounted for by pair wise added particle inte- ractions or the used cell models. In some cases the col- lective motion of a group of interacting particles can, for small deformations, be decomposed in a number of normal modes of motion. This is probably also the case for ordered latices. To each normal mode of mo- tion a relaxation time can be assigned. The number of modes will be high considering the number of particles involved. The longer relaxation times are pertinent to relaxation of coherent mode of motion of particles ov- er a long distance and the shorter times are pertinent to those over short distances. The shortest being those of pairs of particles. This qualitative picture allow a few predictions. For sufficient high frequency the shear forces dominate any finite restoring force. Consequently one expects that at high frequency G' and q' levels off as function of the frequency, because of the finite number of relaxation times. The lim G~,_ co is the sum of contributions of all the relaxation mecha- nisms including those belonging to independent pair interactions. The viscosity lim q~,_ oo is equal to the viscosity of only hydrodynamically interacting particles. For hard spheres this limit is in good approximation given by the semi-empirical formula of Simha [17]: 10 9 R 3 (R 7 - 1) r/'~el = 4 (R 1~ + 1) - 25 (/7 + R 3) + 42 R 5 (10) where R = f q~- ~/~. (11) Empirically it is found that the hydrodynamic inte- raction factor jr = 1.11 for electrical neutral particles and q) < 0.15 [18]. This value of/ is  used to calculate the in figure 2 indicated limits. According to Simha [17] the value of jr will be higher for a system with electrostatic interaction, which results in lower values of r/'re I. This explains probably the difference between the mea- sured and calculated limit values of r/',e I for c0--+ oo (see figure 2). Another prediction which can be made is that at a fixed frequency one expects G' and r/' to decrease with increasing electrolyte concentra- tion, because the characteristic length of long distance ordering gradually decreases by the addition of elec- trolyte so that the longest relaxation mechanisms are annihilated. This can be observed in figure 3 and 4. Summarizing we condude that G* of (partly) ordered latex is strongly frequency dependent and can only be described with a great number of relaxation times. There is an indication that long range ordering is bro- ken down by adding of electrolyte. It seems necessary to develop temporary network theories to analyse the observed phenomena adequately. References 1. GoodwinJW, Khidher AM (1967) Colloid and Surface Science, ed Kerker, Academic Press New York, Vol 4, p 529 2. Mitaku S, Ohtsuki T, Enari K, Kishimoto A, Okano K (1978) Jap J of Appl Phys 17:305 3. Ohtsuki T, Mitaku S, Okano K (1978) Jap J of Appl Phys 17:627 4. Mitaku S, Ohtsuki T, Kishimoto A, Okano K (1980) Biophys Chem 11:411 5. Russel WB, Benzing DW (1981) J Coil and Int Sci 83:163 6. Benzing DW, Russel WB (1981) J Coil and Int Sci 83:178 7. Buscall R, Goodwin JW, Hawkins MW, Ottewill RH (1982)J Chem Soc Far Trans I 78:2873 8. Buscall R, Goodwin JW, Hawkins MW, Ottewill RH (1982) J Chem Soc Far Trans I 78:2889 402 Colloid and Polymer Science, VoL 262. No. 5 (1984) 9. Lindsay HM, Chaikin PM (1982) J Chem Phys 76:3774 10. Strivens TA (1983) Colloid & Polymer Sci 261:74 11. Norde W (1976) Proteins at interfaces, mededelingen Land- bouw hogeschool Wageningen, Nederland 76-6 12. Blom C, Mellema J, accepted for publication in Rheol Acta 13. Mason WP (1964) Physical Acoustics, Academic Press New York, vol I-A, chap 1 14. Te Nijenhuis K (1979) Thesis Delft 15. Williams R, Crandall RS, Wojtowicz PJ (1976) Phys Rev Let- ters 37:348 16. Flfigge W (1975) Viscoelasticity, Springer-Verlag, Berlin, Hei- delberg, New York 17. Simha R (1952) J Appl Phys 23:1020 18, Thomas DG (1965) J Colloid Sci 20:267 Authors' address: C. Blom Department of Applied Physics Twente University of Technology P. O. Box 217 7500 Enschede, The Netherlands Received November 2, 1983; accepted December 28, 1983 

Frequency dependent linear viscoelastic properties of ordered polystyrene latices

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