Langmuir 1992,8, 2863-2872
2863
Deformation, Melting, and Relaxation of Structured Colloidal Dispersions A. Bradbury, J. W. Goodwin,+and R. W. Hughes School of Chemistry, University of Bristol, Cantock's Close, Bristol BS8 ITS,U.K.
Received March 16,1992.In Final Form: June 1, 1992 Strongly interacting dispersions of sphericalparticles of silica and poly(viny1idene fluoride) have been prepared at high concentrations. The linear and nonlinear viscoelastic response of these systems have been investigated. These systems show a strong elastic response characteristic of viscoelastic "solidlike" behavior. It is demonstratedthat by calculatingthe dimensionlessinternalenergy density,a phase diagram can be constructed for the onset of viscoelastic solidlike behavior. The linear viscoelastic response has been calculated using a model for the relaxation spectrum which gives excellent agreement with the data. A model is proposed for the nonlinear viscoelastic response for small to moderate strains. It is based on the idea that increasing strain causes progressive melting of the order in the dispersion into a metastable liquid phase. It is demonstrated that when a scaled strain or an equivalent temperatureis plotted against the fraction of liquid present, the nonlinear viscoelastic response of these systems can be reduced onto a master curve. 1. Introduction
The onset of phase changes in colloidal systems is accompanied by changes in the bulk material properties. Measurements of the bulk properties of a system which leave it unperturbed are most easily related to the microstructural order; such tests are nondestructive experimenta (NDEs). In appropriate conditions the scattering of both light and neutrons by dispersionscan provide a description of the microstructure and short time dynamics of a system; at low radiation levels these are NDEs. Mechanical spectroscopic tests provide a direct measure of the bulk behavior, and long time dynamics for a wide range of systems and at low strains and stresses are also NDEs. Computer simulations aid understanding of both the microstructural order and the material properties of a system nonintrusively. Simulations of the osmotic pressure's2 in hard sphere systems have shown that with increasing concentration the pressure of the system rises with volume fraction and the system maintains a liquidlike order. At a critical volume fraction a further increase in concentration over a small volume fraction range results in a constant osmotic pressure. In this region there is a coexistencebetween geometricallyordered and disordered states. As the system becomes dominated by the growth of the ordered state, the osmotic pressure begins to rise again. Visual inspection of near hard sphere systems by Pusey3 illustrated that as the concentration of the particles is increased, they form Bragg reflecting crystallites coexisting w i t h a less well ordered state. The onset of crystallization was used to determine the volume fraction 4, of the system in order to agree with the computer simulation of Hoover and Ree.' The value of the volume fraction at freezing was 4 = 0.494. A t an effective hard sphere volume fraction 4 = 0.536,the system was composed almost entirely of crystallites. As the volume fraction was increased toward random close packing of 0.63, an amorphous glassystateformed. For an ideal hard sphere system at thermodynamicequilibrium,the internal energy of each phase either side of the order-disorder transition can be consideredin terms of the particlesuncoupledtranslational (1) Hoover, W. G.; Ree, F. H. J. Chem. Phys 1968,49, 3609. (2) Woodcock, L. V. Ann. N.Y. Acad. Sci. 1981,37, 274. (3) Puaey, P. N.; Van Megen, W. Colloidal Crystals and Glasses. In Scientific Methods for the Study of Polymer Colloids and Their App1ications;NATOASISeriee;Candau,F.,Ottewill,R.H.,Eds.;Kluwer, Boston, MA, 1990; Vol. 303, pp 413-426.
motion. The energy of a particle undergoing Brownian movement is 3kTl2 and in hard sphere systems there is no pair interaction contribution except at contact. If there is no change in the kinetic energy contribution, there is no internal energy change for the transition. The change in the Helmholtzfree energy for the transition is dominated by the entropy. These systems are unlikely to be appropriate models for melting transitions in alloys where atoms interacting through strongly repulsive potentials show profound changes in the pair interaction energies on melting. For near hard sphere systems at large volume fractions, glassy states form. These can be considered to be metastable not representing the lowest energy configuration; the lifetime of this state is often much longer than that of the observation time. For strongly interacting particles such as polystyrene latex in solutions of low concentrations of electrolyte, the formation of crystallites occurs at very much lower volume fractions than for hard spheres. Initially, body centered cubic (bcc) crystallites form and as the concentration is increased; face centered cubic (fcc) structures dominate? A further increase in the volume fraction results in the formationof finer grains tending toward a glass-likephase. The Helmholtz free energy change for charged systems must include a significant contribution from the internal energy as this is controlled by the pair interaction energy. The time scale for changea in these systemscan take several minutes to many days and the structures will be conditioned in part by the deformation history of the material. For such systemsa linear viscoelastic measurement ( W E ) is vital in understanding the temporal evolution of the order. Nonlinear measurements are important in understanding what role deformation history has played in conditioningthe samples. An understanding of nonlinear viscoelasticityis also vital in aiding the practical utilization of concentrated dispersions. In the following sections the calculated internal energy for strongly interacting systems will be compared with data for the onset of viscoelastic behavior. A range of dispersions which show elastic behavior is investigated as a function of strain and applied frequencies. A model is developed which reduces the data onto a master curve. (4) Williams, R.;Crandall, R. S. Phys.Lett. A 1974,48,225. Goodwin, J. W.; Ottewill, R. H.; Parentich, A. J. Phys. Chem. 1980,86,5127.
0743-746319212408-2863$03.00/0 0 1992 American Chemical Society
2864 Langmuir, Vol. 8, No.12, 1002
Bradbury et al.
These data can be interpreted in terms of an equivalent melting temperature and the degree of liquid-solid coexistence as the strain is increased. 2. Background 2.1. Viscoelasticity. The linear viscoelastic response
of a material can be described in terms of two major classes of phenomenological behavior: a viscoelastic solid or a viscoelastic liquid respon~e.~ The response of a linear viscoelastic solid can be visualized in terms of the application of a step strain. The material instantly respondswith a step increase in stress. The stress divided by the strain defines the relaxation function or modulus, G ( t ) . If the material were purely elastic, the stress would be maintained at a constant value while the strain is applied. A linear viscoelastic solid may relax this strain to some extent with time but the stress and hence the relaxation function will never reduce to zero. A linear viscoelastic liquid will respond to a step strain with a step stress, but this stress can be completelyrelaxed. It is convenient to describe the relaxation function in terms of a distribution of uncoupled mechanical dipoles which relax independently with a characteristic time T . The value of each dipolar contribution is given by the relaxation spectrum, H(T).By summing all the spectral contributions at an experimental time t, the relaxation function can be calculated.
The vectorial sum of these terms gives the complex modulus G*(w) where
+
G*(o) = G’(u) iG”(o) (5) where i2 = 1and w is the applied frequency in radls. The application of an oscillating stress gives rise to an oscillating strain which can also be divided into an inphase and out-of-phase strain. The maximum strain divided by the maximum stress defines the complex compliance J*(o).The complex modulus is the reciprocal of the complex compliance G*(w) = l/J*(w) (6) where the complex compliance is related to the storage compliance S(U)and the loss compliance S’(U) as J*(U) = J’(w) + i S ’ ( U ) (7) By substitution of eqs 5 and 7 into eq 6, the relationship between the storage and loss compliance with the storage and loss moduli is obtained
S(w) =
G ( t ) = c m e - t ’ TH(T ) d In ( 7 )
As the experimentaltime t
(1)
0, the above integralbecomes
+
G’(w) ( G ’ ( u ) ) ~ (G”(w))’
+
G’’(U) (G’(o))’ (G”(w))’ The corresponding integral equations for the storage and loss compliance are S’(w) =
+
where G(-) is the high frequency limit to the shear modulus. The initial value of the relaxation function for either a viscoelastic solid or liquid gives the high frequency limit to the shear modulus. The zero frequency or low shear rate viscosity ~(0) can be obtained from an integral of the spectrum
For a viscoelastic liquid the integrand must converge to zero with increasing relaxation time, whereas for a viscoelasticsolid this integral need not be finite and so the material does not possess a zero frequency or zero shear rate viscosity. An alternative representation of a viscoelastic solid is to add a static modulus G(0) to the relaxation function and the storage modulus (eqs 1 and 2). If the static modulus is nonzero, the material is a viscoelastic solid; if it is zero, it is a viscoelastic liquid. A static modulus can be represented as a Dirac delta function at infinite time in the relaxation spectrum. The application of an oscillating strain gives rise to an oscillatingstress which can have components in phase with the applied strain and a quadrature component. The stress which is in phase with the applied strain divided by the maximum strain is termed the storage modulus G’(u) and the stress which is out of phase with the applied strain divided by the maximum strain is the loss modulus G”(u). The frequency dependence of these moduli can be obtained from the Fourier transform of the stress relaxation function6 (5) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (6) Gross, B. Mathematical Structures of the Theories of Linear Viscoelasticity; Hermann: Paris, 1968.
The term U T is ) the retardation spectrum and J ( = )is the infinite frequency compliance. It has been observed in viscoelastic experimente that systems of strongly interacting particles showa viscoelastic liquid response over a very small volume fraction range.’* For many latex systems as the high frequency modulus rapidly increases with volume fraction the zero ehear rate viscosity rapidly becomes large and very difficult to establish.’-” These systems show viscoelastic solid behavior during the measurement time scale, typically of a few milliseconds to a few hours. Therefore the product of the apparently infinite zero frequency viscosity and the frequency will make only a negligible contribution to the loss compliance. The loss compliance becomes dominated by the retardation process as would be expected from a viscoelastic solid. The limit of the linear viscoelastic response is often tested by increasing the strain amplitude at a f i e d frequency and recording the strain where the modulus is no longer independent of the strain. For structured colloidal fluids this represents the point where the (7) Buscall, R.; Goodwin, J. W.; Hawkins, M.; Ottewill, R. H. J. Chem. SOC.,Faraday R a m . 1 1982, 78, 2873, 2889.
(8) Goodwin, J. W.; Hughes, R. W. Mater. Res. Soc. Symp. h o c . 1990, 177, 187. (9) Keeping, S. A. PhD Thesis, University of Bristol, 1989. (10) Goodwin, J. W.; Gregory, T.; Miles, J. A.; Warren, B. C. J. Colloid Interface Sci. 1984, 97, 488. (11) Goodwin, J. W.; Hughes, R. W. J . Chem. Phys. 1991,95, 6124.
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Structured Colloidal Dispersiom
magnitude of the applied strain produces an average particle displacement similar to the mean Brownian displacement. In a thorough viscoelastic study of a near hard sphere system Jones, Leary, and Boger12illustrated that for strains greater than a critical strain, Y d t , the storage modulus begins to reduce. They observed that as the volume fraction is increased,+y&treduces up toa volume fraction of 0.58. Above this volume fraction there is a slight increase in yCdtwith increasingvolume fraction. This dependence might be attributable to some "softness" in the interaction energy between the particles. Zukoski and Chen13have studied the linear and nonlinear viscoelastic response for an amphoteric latex at a wide range of volume fractions and electrolytes. The surface characteristics of their particle were conferred by copolymerization of a polyelectrolyte with insoluble polymer, and so the surface of their particles will be more diffuse than systems described in section 3. They observed that there are strong similarities between their ordered systems and found a critical strain that was relatively insensitive to the electrolyte concentration. Using a model based on a sinusoidallyvarying potential energy for the slip of planes of particles they were able to calculate a critical strain re, where slip occurred. This value was 0.04 which was in reasonable agreement with their experimental data. A similar calculation for metals gave a value of 0.16,14 although incorporating factors such as dislocations and the form of the potential reduces this value to 0.03. The important feature of the calculation is that once the pair potential is large enough to cause a transition from liquidlike order, the critical strain is dependent on the form of interparticle interaction rather than its magnitude. The interparticle potential also controls the internal energy of strongly interacting viscoelastic dispersions. The link between the thermodynamic properties and the viscoelastic properties can be established by considering the relationship between the internal energy and the elasticity of the solid. 2.2. Internal Energy. The internal energy of the particles for a system interacting through a strongly repulsive potential can be considered as consisting of two terms: firstly, a kinetic energy term for the translational motion of the particles which is proportional to 3kT/2 where kT is the thermal energy; secondly, the configurational potential energy associatedwith the order induced by the pair interaction energy between the particles. Consider a sphericalshell surrounding a particle at a radial distance r from the particle and with a thickness dr. The average number of particles in this shell is given by 4u?dr g(r)p, where g(r) is the pair distribution function and p is the average number density. For pairs of particles interacting with a colloid pair potential energy V(r), the total potential energy associated with this shell is 4 d g(r)p V(r) dr. This function needs to be integrated over all r to calculate the potential for a pair of particles and then multiplied by N/2 pairs of particles in the dispersion to give the configurational potential energy contribution. Adding this to the kinetic energy gives the internal energy Vi
-s
3NkT N +Vi = -+ 4ur2g(r)p V(r) dr 2 2 0 The internal energy is proportional to the number of (12) Jones, D. A. R.; Leary, B.; Boger, D.V. J. Colloid Interface Sci. 1991,147,479.
(13) Chen, L.; Zukoski, C. F. J . Chem. SOC.,Faraday Tram. 1 1990, 86, 2629. (14) Kittel, C. Introduction to Solid State Physics, 5th ed.; Wiley: New York, 1976.
particles present in the dispersion and so depends on the volume examined. The internal energy per unit volume of dispersion, E, is thus
For a dispersion in a glassy or fine microcrystalline state it is more appropriate to replace the average number of particles at a distance r by the number of nearest neighbors N1 centered at an FCC separation R 4?rr2g(r)p=) N$(r - R)Iforn-t neighhm (12) where 6(r - R) is the Dirac delta function and R is given by
R = 2a(4m/4)1/3
(13) and a is the particle radius. If fcc geometry is maintained on a local scale, NI = 12 nearest neighbors and the maximum packing fraction tpm= 0.74. By use of expression 12 in eq 11, the internal energy per unit volume of the dispersion becomes
E
+
3pkT/2 3NlV(R)c$/8ua3 or in a dimensionless form
(14)
where E = Ea3/kT. The high frequency limit to the shear modulus is also related to the pair potential between the particles7
Substituting into eq 16 for the internal energy per unit volume of dispersion, the following result is obtained
This expression is best applied to strongly interacting systems. The volume fraction dependence of the high frequency limit to the shear modulus is well-known for a number of these systems7-10and the parameters used to calculate G(-)can be used to determine the dimensionleaa internal energy per unit volume of a dispersion. These data are plotted in Figure 1 as a function of volume fraction. This plot contains data for three different latex materials, polystyrene, silica and poly(vinylidenefluoride). It covers a range of radii from 26 to 150 nm, surface potentials from 40 to 90 mV, and electrolyte concentrations from 1W2to 5 X lo4 M. The experimentally determined boundary between the viscoelastic solid response and the viscoelastic liquid shown in Figure lis given by the empirical function
E = 0.05/4 for 4 I (2u/45)'l2
(18) This indicates that as the volume fraction is reduced, the energy density required to form a viscoelastic solid increases. This curve intersects the hard sphere internal energy at a volume fraction where the internal energy density is equal to ( 2 ~ / 4 5 ) ' / ~A. system with an energy density less than this or below the hard sphere internal energy of 3pkTl2 is either weakly attractive or coagulated. Experimentally the viscoelastic solid phase boundary is recognized by a rapidly increasing viscosity. It should be noted however that this behavior is often associated with the formation of glass phases. The volume fractions of the dispersions studied below have their internal energy
Bradbury et al.
2866 Langmuir, Vol. 8,No. 12, 1992
100
:
-
0
10
e
0.1
si
C
c
t
I
E
0.01
J
'
0 001 0.1
0.2
I
1
0.3
0.4
I 0.5
0.6
Volumr Iraction
Figure 1. A plot of the reduced internal energy density as a function of volume fraction. The experimental data points indicate systems which demonstrate 1-liquid behavior, 1 s viscoelasticliquid behavior and s viscoelasticsolid behavior. The upper solid line indicates the boundary between viscoelastic solid and liquid behavior. The lower line shows the internal energy density of a hard sphere system 3pkT/2. Key: 0,polystyrene, 34.3 nm radius, 5 X 10-1mol dm-3 electrolyte concentration, 50 mV surface potential; 0 ,polystyrene, 26.3 nm, 1X mol dm" 50 mV; 0, polystyrene, 34.3 nm, 1 X mol dm-3 55 mV; X, polystyrene, 39.2 nm, 1 X lC3mol dm-3 50 mV; +, polystyrene, 98.3 nm, 1 x 10-3 mol dm-3 89 mV; A, polystyrene, 85 nm, 5 x lo-' mol dm-3 40 mV; 0, PVDF, 110 nm, 2 X 10" mol dm-3 80 mV; H, polystyrene, 37.6 nm, 1 X lC3 mol dm-3 45 mV; e, polystyrene, 37.6 nm, 1X 10-1mol dm-3 40 mV; A, polystyrene, 37.6 nm, 1 X 10-2 mol dm-3 45 mV; H, silica, 150 nm, 1 X l W 3 mol dm-3 80 mV.
+
density greater than that required to produce a strongly viscoelastic solid phase. 3. Experimental Section 3.1. Characterization and Preparation of the Systems.
The colloidal silica sample was prepared by the method outlined by Stober et al.lS and Zukoskile, with tetraethylorthosilicate (B.D.H. GPR grade), ammonia (B.D.H. 880 GPR grade), and ethanol (B.D.H. GPR grade) being used as supplied. The silica dispersions were extensively dialyzed before use. It was particle sized by transmission electron microscopy and found to have a mean particle radius of 150 nm with a coefficient of variation of 5 %, The silica samples studied in lO-9 M sodium chloride were dialyzed against this concentration of electrolyte at pH 6 prior to concentrating by centrifugationto the required volume fraction. The silica samples studied in 5 X lo-' M sodium chloride were dialyzed against water supplied by a Purite Labwater system. The water was adjusted to pH 6. The samples were then prepared by repeated centrifugation and redispersion in solutions of 5 X 104M sodium chloride solution. The conductivity of the system was monitored and the washing procedure was terminated once the conductivity of the supernatant reached the value of the added electrolyte. The latex of poly(viny1idene fluoride) (PVDF), supplied by Dr. R. Buscall (IC1 Runcorn), was dialyzed against an aqueous solution of sodium hydroxide at pH 8 in water supplied by a Purite lab water system. It was coated with a monolayer of the surfactant hexaethylene glycol monododecyl ether obtained from Nikko chemicale (Tokyo, Japan). This has a geometric chain length of 3.85 nm. Transmission electron microscopy was used to size the uncoated particles which were found to have a radius of 106 nm and a coefficient of variation 3%. Concentrated samples were prepared by repeatedly centrifuging and redisM BDH AnalaR grade sodium persing in solutions of 2 X chloride at pH 5.7. (15) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968,26, 62.
(16) Bogush, G. H.; Tracy, M.A.; Zukoeki, C. F. J. Non-Cryst. Solids 1988,104,95.
01
0 01
1
Slraln
Figure 2. Strain sweeps for PVDF as a function of volume fraction measured at 5 Hz: 0 storage modulus, 0 loss modulus, 0.451 volume fraction; W, 0, 0.420; e, 0,0.383; A, A, 0.341. Electrophoretic mobility of these systems was measured using a PenKem system 3000. The mobility of PVDF at pH 5.7 in an M sodium chloride was 3.54 electrolyte concentration of 2 X X 10" (m s-l)/(V m-l). The calculated zeta potential was 68 mV.17J8 The electrophoretic mobilities of silica were measured at pH 6.3 and at 1X lop3M sodium chloride gave a value of 4.30 x 10-8 (m s-l)/(V m-9 and at 5 x 10-1 M sodium chloride gave a value of 5.00 X 10" (m s-l)/(V m-9. This was converted to zeta potential and gave 67 and 83 mV, re~pectively.~~J~ 3.2. Rheological Measurements. Rheological experiments were carried out using Bohlin VOR and CS rheometers. The VOR instrument was used to apply known strains or strain rates and the CS instrument to apply known stresses or stress rates. Three designs of instrument geometries were used in this study: a cone and plate; a double concentric cylinder; a bob and cup (single concentric cylinders). The geometries were sealed with a thermostated environmental housing which maintained the temperature and humidity of the environment above the samples. This was found to prevent significant solvent evaporation from the samples for periods of several hours depending upon the volume fraction. The strain and stress dependence of the samples was determined using an oscillating strain and stress sweeps. In this mode the Bohlin rheometers generate strains or stresses with increasing amplitudes. These experiments were performed at a number of frequencies, typically in the range of 0.1-10 Hz. The typical to 10-l and the typical stress strain range examined was range was to 10 Pa. The strain sweeps for a range of volume fractions are shown in Figure 2 for PVDF and Figure 3 for silica dispersions. Also shown in Figure 4 is a typical stress sweep for a silica dispersion. Both types of experiments showed the system to be a viscoelastic solid exhibiting a linear regime up to a critical point where G* or J* were independent of the applied strain/stress. A t this point the system became increasingly fluid with the storage and loss components eventually crossing over. The complex compliance and modulus display only a very weak frequency dependence with an applied strain in the linear regime. This was confirmed by an oscillation experiment (a frequency range of 10 to 0.005 Hz was covered) where G* or J* were found to be linear with frequency. A second oscillation experiment (from high to low frequency) was carried out followed by a suitable delay time prior to a stress relaxation experiment being carried out. This was to remove as much of the shear history as practicable. A near step strain was applied, this rose to ita maximum value in 0.2 8. The strain was applied in the linear regime, the stress response only decayed very slowly and showed no evidence of reaching a constant value after 6 h. The response of silica is shown in Figure 5. This is indicative of a viscoelasticsolid. Similar behavior is shown by PVDF in Figure 6. The strain was increased beyond the linear region and a series of stress relaxation experiments were performed. A relaxation curve as a function of applied strain is shown in Figure 7. It was (17) O'Brien, R. W. and White, L. R. J. Chem. Soc., Faraday Trans. 2 1982, 74, 1607.
(18) Henry, D. C. Proc.
R. SOC. Ser. A
1931,133, 106.
Langmuir, Vol. 8, No.12,1992 2867
Structured Colloidal Dispersions 500
,.."A
B
a
400
100
300 10
I
0.1
0.001 0 01
0 001
01
1
strain
.,
Figure 3. Strain sweeps for silica in 10-8 M sodium chloride as a function of volume fraction. The data were gathered at 1 Hz. Key: e storage modulus, 0 loss modulus, 0.455volume fraction; 0 , 0.431;A, A, 0.378;e, 0 , 0.3%; X,
I
I
I
I
I
1
1000
10
0.1
Ilme I
Figure 6. Stressrelaxation function for PVDF in the low strain limit.
+, 0.340.
200
-
150
-
100
-
50
-
B c
1
I'
" 20
\::
J'
e ...
c c
0.05 0.04
a
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B
0 01
0 001
01
slraln
I
I
0.1
I
10
0 001
I I
I
10
0.1
, 1000
s
Figure 7. Stress relaxation function for PVDF as a function of the applied strain. 0,0.0104;0,0.0207;0,0.0414;X, 0.0622;+,
1
Figure 4. A stress sweep and a calculated strain sweep for silica C#I = 0.331in 5 X lo4 M sodium chloride. I
o r
time
0
0.001
-
40.02
- 0.01
1000
=6
I
1000
time
Figure 5. Stress relaxation function for silica in M sodium chloride. found that the stress relaxation curve decays faster for higher strains. In the experimental strain range of the geometries used in this study it was found that the stress relaxation curve did not reduce to zero.
4. Discussion The above rheological characterization includes both NDEs and nonlinear tests. At low strains there is no significant perturbation of the order in the system and its response can be modeled using arguments based on slow density fluctuations in the material. 4.1. Calculationof the Stress RelaxationResponse. A model that describes the relationship between the
0.0829;A, 0.104; *, 0.145;0 , 0.176; 0 , 0.207.
average microstructural order in a simple molecular liquid and the linear viscoelastic response of its high frequency limiting shear modulus was derived by Zwanzig and Mountain.lg This approach has been successfullyused to describe weakly attractive colloidal systems.20The linear viscoelasticresponse of strongly repulsivecolloidalsystems has been well described by eq 16.7-11The shear moduli of the silica samples are only a weak function of frequency and so it can be assumed that these values represent the high frequency limit to the storage modulus. The lattice model is used to fit to shear modulus as a function of volume fraction (Figure 8). The best fit was obtained by adjusting the value of the surfacepotential on the particles in the model. The fitted value in a concentrated system compares favorable with electrophoresis data for the zeta potential gathered on low volume fraction samples. The stress relaxation behavior of a particular volume fraction of a latex such as silica or PVDF can be obtained from the calculated relaxation spectrum." In this approach the dispersion was modeled as a series of regions containing different particle number densities. These states were considered to be fluctuations from the a reference state with an equilibrium separation R,which is the average separation of a nearest neighbor to a test particle. This separation was used in a cell model (eq 16) to calculate G(-) as a function of volume fraction. This was found to compare well with experimental data. The magnitude of the relaxation spectrum is considered to be a measure of the change in elasticity of the regions (19) Zwanzig, R. W.; Mountain, R. D. J. Chem. Phys. 1965,43,4464. (20) Goodwin, J. W.; Hughes, R. W.; Partridge, S. J.; Zukoeki, C. F. J. Chem. Phys. 1986,85,559.
Bradbury et al.
2868 Langmuir, Vol. 8, No. 12, 1992 h
-a
1500
1
I
I
I '
I
1000 L
I
I
I
I
I
101
I
I
I
I
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i f
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5
500
a 0
P 0 03
0 34
0 38
0 42
0 46
05
Volume fractlon
0.001
I
0.1
10 the I
1000
Figure 8. Measured high frequencylimit to the shear modulus for PVDF, polystyrene, and silica systems compared with the calculated response using fitted surface potentials J..
Figure 9. Stress relaxation function for PVDF in the low strain limit compared with the model.
due to a fluctuation in density. The accompanying fluctuation modulus AG(T),occurs with a characteristic time T and a probability P(V,
a pair of particles are displaced from the reference state, the interaction energy between them will rise, slowing the particle motion. The difference in energy U, between the reference state and the new state can be used to define an activation energy barrier hindering the diffusive process
H(T)= AG(T)P(U) (19) The probability of a fluctuation occurring was given by the Boltzmann equation. The energy required to achieve a particular configuration was given by the change in the colloid pair potential energy U from a reference state with a nearest neighbor separation R to a new state with a separation R(7). The colloid pair potential V(R) was calculated from the model proposed by Derjaguin,Landau, Verwey, and Overbeek (DLV0),21from which the energy increase, U , above the reference state can be obtained
U V(R(T))- V(R) (20) It is important to recognize that the local concentration changes are slow enough to ensure that on a typical experimental time scale it is likely that the temporally and spatially averaged density distributions will differ. The system is nonergodic. In order to obtain the value of the probability P( v), a knowledge of all the configurations present in the dispersion is required. In part this will depend on the history of the sample and is best obtained by ensuring that the time integral of eq 19 is equal to the measured G(-), which is a requirement of linear viscoelastic theory. This results in a constant, A, related to the sum of all those states present in the dispersion. In this manner the experimental system is used to define the partition function. In summarythe value of the relaxation spectrum at the time, T , is given by the product of the fluctuation in modulus with a Boltzmann probability of the fluctuations occurrence
All of the above terms can be calculated from the parameters which determine the DLVO curves. The time required to achieve a fluctuation is related to the collective diffusive motion of the structure. The time T can be evaluated from a time-dependent diffusion coefficient D(T)and the mean square distance diffused = ((R - R(T))~)/D(T)
(22) where the angular bracketa denote the average value. As T
(21) Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids Elsevier: New York, 1948. b r i n k , H.; Overbeek, J. Th. G. Discus. Faraday Sac. 1964,18,14. Derjaguin, B. V.; Landau, L. Acta Physichim. USSR 1941, 14, 633.
= D~e+' (23) where B = (kZ")-l, k is the Boltzmann constant, T is the temperature, and Di is a long time diffusion coefficient of the reference state. The above expression describes the change in the diffusivebehavior for a particular fluctuation with an energy V; as the energy depends on the distance moved, the diffusion term D ( T )is not a simple exponential. The slow diffusive motion in bcc crystal structures has been measured by Asher and co-workers.22 In an innovative experiment they were able to use high intensity laser light to displace dyed latex particles (4 = 0.0205) from an equilibrium configuration into a denser structure. The relaxation of the structure followed an exponential and the characterization relaxation time of the structure was of the order of 5 s. For higher concentration systems in which fcc structures dominate, structural relaxation will be very slow and a spectrum of behavior is more likely. In a previous publication'l it has been shown that the diffusion coefficient Di can be estimated from the Lindemann melting rule.23 This assumes that once a particle had diffused a distance equal to about '/loth of the lattice separation, the material has melted. The characteristic , assumed to occur when the time for this melting, T ~was relaxation function reduces to l/e of ita original value. This can be visualized as a lattice strain y1, where ~ ( 7 )
AR = YlR
(24) where AR is the displacement of a particle from ita equilibrium position R. For a Lindemann strain this is y1 = 0.1. With the appropriate substitutions and rearrangementa of eq 22
Di = ((RyJ2)eBU/Tc (25) and the energy U is given by the difference in the pair interaction energy at R and R + AR
U = V(R + AR)- V(R) (26) Figure 9 shows the measured relaxation function for three volume fractions of the latex of PVDF. The calculated curve was obtained by integration of the (22) Rudquiat, P. A.; Keeevamoorthy, R.; Jagannathan, S.; Asher, S. A. J. Chem. Phys. 1991,96,8546. (23) Lindemann, F. A. Phys. 2. 1910, 11, 609.
Langmuir, Vol. 8, No. 12, 1982 2869
Structured Colloidal Dispersions
eo
b 40
;
s
t
f
1
100
1 o4
1 0’
1 0’
lllm 8
Figure 10. Calculated relaxation spectra for PVDF. calculated relaxation spectra shown in Figure 10. The calculation was performed by obtaining a trial relaxation spectrum for 71 = 0.1. This was integrated to calculate the relaxation function. The agreement with the experimental data was improved by iteratively adjusting the value of 71. This is the only adjustable parameter in the model, and the best fit was obtained with strains of 0.070.08. This strain represents the point at which the reference state will melt, which has a nearest neighbor separation of an fcc crystal. The calculations of Mitchell, Ninham, and c o - ~ o r k e r ssuggest ~ ~ ~ ~that ~ for systems interacting with a potential of the form r-” the melting strain is insensitive to n and for values of n = 4-12 the strain yl varies from 0.100 to 0.106. This response reflects the insensitivity of the first peak in the structure factor to the value of n in this range of values. However in order to obtain agreement with computer simulations of Coulombic interactions, the melting strain was found toreduce and take values between 0.07 and 0.08, which is in excellent agreement with the model presented here. 4.2. A Model for Nonlinear Viscoelastic Response. The above model for the relaxation spectrum provides a good description of experimental data for an NDE. In the following sections the nonlinear viscoelastic response of a viscoelastic solid with the observed slow relaxation behavior is considered. The approach adopted leads to the concept of a reduced temperature for the dispersion related to the applied strain and a change from solidlike to liquidlike order. The model is based on four basic assumptions. 4.3. Assumptions. The physical assumptions used in the microstructural description of the nonlinear viscoelastic response of monodisperse spherical particulates are as follows: (i) The material consists of a fine polycrystallineor glassy structure in which the number density fluctuations occur on a long time scale. Typically a few millisecondsto several hours. (ii) In the low stress or strain limit the material is modeled as being composed of a series of grain each with a viscoelastic solidlike response. (iii) The total compliance of the dispersion is the s u m of the compliance of each individual grain. (iv) As the strain is increased, those regions in the dispersion with a relatively low number density begin to distort from a regular fcc geometry and assume a more (24) Forsyth, P. A.; Marcelja, S.;Mitchell, D. J.; Ninham, B.W.Adu. Colloid Interface Sci. 1978,9, 37. (25) Marcelja, S.; Mitchell, D. J.; Ninham, B.W.Chem. Phya. Lett. 1976, 43, 353.
liquidlike order. This dynamically induced phase has a viscoelastic liquid response but the product of the zero shear viscosity and frequency is large enough to provide only a small contribution to the loss compliance J” of the liquid. The model is best applied in regions where the nonlinear stress or strain does not result in significant flow. 4.4. A Model for Strain Induced Melting. Consider the strain sweepsobtained for PVDF and silica dispersions in Figures 2 and 3. It was observed that as the strain is increased, the storage modulus reduces and the lose modulus increases to a maximum value and begins to reduce with further increases in strain. The shapes of these curves are typical for a wide range of size and salt concentrations for systems showing behavior typical of viscoelastic solids. In the following model the reduction in the storage modulus is considered to reflect the lose of solidlike order in the dispersion and the lose modulus grows to reflect the increase in liquidlike order within the dispersion. In the small strain limit the storage and the loss compliance can be calculated either from a direct measure of the oscillatingstrain from an applied oscillating stress or from the determination of resulting stress from an applied oscillating strain using the storage and lose moduli in eq 8. The total measured storage compliance JT’(~) and loss compliance J T ” (can ~ )be calculated from a linear superposition of compliances for the liquid and solid regions JT’(7)
= &(y)J,’
+ 4L(y)JL’ + h(7) JL”
(27)
JT”(y) = $S(Y)JS” (28) where J’ is the storage compliance, J” is the loss compliance, the subscript T represents the total measured compliance, S the compliance due to grains with a viscoelastic solid configuration, and L the compliance of grains with a metastable liquid configuration. The parentheses indicate that these terms are functions of strain y. The fraction of regions with a particular configuration is given by &,(y) and &(TI. These values are chosen such that
+ 4&)
=1 (29) The stress and strain sweeps show no strong frequency dependence so that in the frequency range examined the viscous contribution to the loss compliance is small. An inspection of eq 9 indicates that the viscosity of the metastable liquid is very large for this condition. In the small strainlstress limit where virtually all the regions possess a solidlike order, than eqs 27 and 28 reduce 4L(Y)
to J ~ ’ ( y 4 =) J,’
(30)
J ~ ” ( y 4= ) Js”
(31)
since &~(y) = 0 and 4s(y) = 1 as the strain becomes vanishingly small. These values can be obtainted directly from the low stress limit of the storage and loss compliance from a stress sweep. Altematively the compliances can be obtained from the storage and loss modulus in the low strain limit using eq 8. For viscoelastic solids in the low strain limit as the high frequency limit is approachedthe storagemodulus is much greater than the loss modulus. In this limit the highfrequency stress response is related to the strain response by the high frequency limit of eq 6 J*(=) = J’(=) = l/G(m) (32) So substitutingfor the shear modulus in eq 32 and setting
Bradbury et al.
2870 Langmuir, Vol. 8, No. 12,1992
it equal to eq 30, the following result is obtained
In order to calculate the volume fraction of the liquid present in the dispersion, the storage and loss compliance of the metastable liquid are required. If it is supposed that once the applied strain has caused half the regions in the dispersion to melt to a metastable liquid, then 4 ~ (YM) = &S(YM) = 0.5 where YM is the lattice strain where the material melts. The material response at this strain will be equally viscous and elastic so that this strain, YM, will occur when the storage is equal to the loss compliance and also the storage and loss moduli are equal. At this strain eq 27 and 28 can be used to calculatethe appropriate compliances of the metastable liquid phase WT’(YM) = JT’(y+) WT”(yM)
= JT”(y-4)
+ JL’
(34)
+JL”
(35)
The reduced strain can be regarded as a reduced temperature, so if TMis the melting temperature and T the apparent temperature of the solidfliquid, then the reduced strain can be written
T fT M = y f y M
(36)
A simple rearrangement of eqs 27 and 28 can now be used to express the strain-dependent volume fraction of metastable liquid in terms of the measured compliance
1-
JT’(r)J,’’
&L(y)=
fraction samples will show an increasingly frequencydependent compliance. 4.6. Determination of the Melting Curve. The measured strain sweep has been used to calculate &L(T) the volume fraction of metastable liquid present. It was found that allthe data reduced on to a master curve. These data are shown in Figure 12, and Table I gives a list of the critical melting strains for the dispersions. As the volume fraction increases, the melting strain tends to increase. Also included are some data on polystyrene latices. The moat notable deviation occurred for the dispersion of small polystyrene particles which were more brittle than either the silica or PVDF dispersions. This could be due to the significant difference in the particle size with respect to the double layer thickness. It should be noted that the response of the polystyrene latex with a volume fraction 4 = 0.343 has an internal energy less than that described by eq 18. This material behaves as a viscoelastic liquid and thus does not conform to the simple model presented in the previous section. These data do not reduce onto the master curve for the reduced temperature dependence versus volume fraction plot shown in Figure 12. Also shown in Figure 13 is the reduced temperature plot for silica which also superimposeson Figure 12. The data can be fitted using the equation
- JT”(r)J,’
(37) r/rM=T/TM
With this expression the measured storage and loss compliance can be used to calculate the volume fraction of metastable liquid present. The behavior described above is represented diagrammatically in Figure 11. The first grain diagram represents the situation where the applied strain is less than yc and the system behaves as a linear viscoelastic solid. The applied strain or stress probes the pair interaction between the particles. There is a distribution of grains with number densities pi in the system giving rise to a spectrum of relaxation and retardation behavior. As the strain is increased to yc, planes of particles at the boundaries of grains slip relative to each other. A plane of particles will move through an harmonic potential environment, and the model developed Zukoski et al.13suggests yc= O(0.04). This process is unlikely to have a thermally induced analogue. Increasing the strain to TM causes the slip zones to modify their structural order and develop into more fluidlike packing. These zones will develop at the expense of the lower number density regions. This mechanism is likely to be similar to a thermally induced process and the strain y1 = O(0.07) will occur between yc and YM. Experimentallyit has been observed that once the strain ~ material is dominated by flow (section reaches 1 . 7 6 7 the 4.5), in this regime the viscosity shows power law behavior as a function of deformationrate giving an index of 0 ( 1 ) . l 1 One limitation of the model presented is that it is only appropriate for materials which show viscoelastic solid behavior or are very viscous liquids at small strains and stresses. It should be noted that as the strain is increased, the fraction of metastable fluid increases. For metastable phases with a low viscosity, the viscous contribution to the measured compliance will become increasingly significant as the strain is increased. Therefore low volume
where N = 21/2and is evaluated from the large strain limit of the plot, X = 0.348 and n = l / 4 fit the power law portion of the curve for the silica and PVDF systems. The coefficients change slightly for the polystyrene system. A major success for the concept of strain melting is that it provides an explanationfor the failure of simpler models, such as time-temperature superpositi~n.~In this approach, higher strains would represent higher temperatures. The relaxation data in Figure 7 would represent an increase in temperature or a shift to shorter relaxation times. The relaxation data shown in Figure 7 should reduce onto a single data curve providingno phase changes have occurred. No such correlation was observed with these data or data for silica and polystyrene samples. Therefore no evidence was found for a simple timetemperature superposition relationship. This reinforces the argument for the model presented here for a phaee change induced by an applied strain. 4.6. Osmotic Pressure,Static Modulus, and Structure Factor. Interestingly, the application of nonlinear strains in stress relaxation experiments indicates some yield behavior. At high strains in Figure 7 it should be noted that relaxation function is tending to a constant value of G(0) 16 Pa. If the Poisson ratio u, is known, the zero frequency bulk modulus can be calculated26
-
(39)
Many fine-grained metals have a ratio of 0.30-0.36. The zero frequency bulk modulus can be equated with the osmotic or internal pressure n, for the dispersion. This can be calculated from a cell model for a solidlike material or from an integral of the pair distribution function for a liquid.8 The cell model gives
where p is the number density and kT the thermal energy. (26)Science Data Book; Tennent, R. M., Ed.; Oliver & Boyd Edinburgh, 1974.
Structured Colloidal Dispersions
Langmuir, Vol. 8, No. 12, 1992 2871
GRAIN DIAGRAMS
F-Fl
At low strains the applied strain probes the interaction potential between a pair of particles. The number densities of the grains varies from grain to grain to give a spectrum of relaxation processes
Y p 3 density 1
Y -’YC Regions of slip
> p>p 2
4
As the strain approaches about 0.04 regions of slip develop at the grain boundaries.
-+&.-eEnergy
There is an harmonic variation in the potential energy between particles as one layer moves with respect to another
Y “Yn As the strain is increased beyond a critical strain the metastable liquid grows at the expense of the less dense region. At the melting strain 50% of the volume is occupied by metastable liquid.
Metastable liquid
Viscoelastic solid
y
