Langmuir 1995,11, 111-118
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The Freezing of Colloidal Suspensions in Confined Spaces J. E. Hug,* Frank van Swol, and C. F. Zukoski Department of Chemical Engineering, University of Illinois, Urbana-Champaign, Urbana, Illinois 61801 Received August 18, 1993. In Final Form: September 23, 1994@ Simulation and experimental techniques have been developed to study the phase properties of a fluid confined between two closely-spaced walls and applied to investigate systems of charge-stabilized colloidal suspensions between repulsive smooth walls in contact with a reservoir. The phase behavior of confined suspensionsis studied as a function of bulk particle volume fraction, surface charge, and wall separation. Complete crystallizationwithin a fxed-sizegap at sufficiently small wall separation occurs at bulk volume fractions well below the bulk freezing volume fraction. The simulations show a strong dependence of the freezing transition on a preferred wall separation corresponding to an integral number of layers. The simulation predictions match experimental observations at wall separations from five to ten particle diameters.
1. Introduction An understanding of the influence of surfaces on the equilibrium and transport properties of bulk fluids is of great importance in advancingfields of adhesion, rheology, adsorption, and tribology. As a result, there have been a growing number of experimental,lj2 t h e ~ r e t i c a l ,and ~,~ computer ~ t u d i e s of ~ -equilibrium ~ and dynamic properties of molecular liquids confined between two solid surfaces. Results suggest that a confined molecular fluid becomes increasingly solid-like as the width of the confining region approaches molecular dimensions. In confined spaces, the fluid's equilibrium structure and dynamic properties are substantially altered f h m those of the bulk. Thus, an accurate description of the behavior of fluids in inhomogeneous systems must take into account boundary conditions and surface-fluid interactions in addition to concepts that describe bulk behavior. The fluid density profile normal to a solid surface oscillates about the bulk density with a periodicity ofabout one molecular diameter close to the surface and then smooths to the bulk density far from the surface. The density oscillations are due, in part, to geometric effects restricting fluid motion and, in part, to surface-fluid attraction, if any. When averaged over a length scale of a few particle diameters, the density typically appears higher at the surface than in the bulk. For a gas below coexistence,one may observe a phase transition at a single solid surface, resulting in formation of a thick but finite layer of liquid at the surface which is in contact with the bulk gas phase.9 Under conditions which correspond to complete wetting at coexistence,the solid surface is covered exclusivelyby the liquid phase such that there is no contact between solid and gas. As two surfaces in a gas approach one another, single wall effects will be seen as long as the two surfaces remain sufficiently far apart. Once the separation is reduced to where the single wall effects overlap, the particles between the surfaces become further Abstract published in Advance ACS Abstracts, December 1, 1994. (1)Israelachvili, J. N.;McGuiggan, P. M.; Homola, A. M. Science 1988,240,189. (2)van Alsten, J.; Granick, S. Tribol. Trans. 1989,33,436. (3)Hansen, J.P.; McDonald,I. R. Theory 0fSimpleliquid.s;Academic Press Inc.: London, 1976. (4)Tarazona, P.;Evans, R. Mol. Phys. 1983,48, 799. ( 5 ) Ma, W.;Banavar, J. J . Chem. Phys. 1992,97, 485. (6)Stevens, M. J.;Robbins, M. 0. Phys. Rev. Lett. 1991,66, 3004. (7)Somers, S. A.; Davis, H. T. J . Chem. Phys. 1992,96, 5389. (8) Magda, J. J.; Tirrell, M.; Davis, H. T. J . Chem. Phys. 1986,83, 1888. (9)Dietrich, S.In Phase Transitions and Critical Phenomena; edited by Domb, C., Lebowitz, J. L., Eds.; AcademicPress, Inc.: London, 1988. @
0743-7463/95/2411-0111$09.00/0
constrained and more highly localized near the walls. At this point a liquid may condense, completely occupying the gap between the surfaces. This is known as capillary condensation. When two surfaces are separated by dense fluid, analogous phenomena may occur. At a single wall one may observe crystallization below coexistence (prefreezing) indicating that at coexistence one has complete wetting by crystal.1° Similarly, when two surfaces are very closely spaced, capillary crystallization may occur when the gap fills entirely with crystal below bulk coexistence. Experimental evidence for the structure of confined fluids has been obtained from studies of the force-distance profiles measured using the surface forces apparatus (SFA1.l Confinement of fluids between surfaces closer than a few molecular diameters induces oscillatory forces interpreted as resulting from the arrangement of the fluid into layers. The periodicity ofthe force profiles is predicted from density profiles developed in computer simulations using hard-spherel1-l4 or Lennard-Jones p o t e n t i a l ~ . ~ J ~ Due to the small separations involved, direct observations of such density profiles have not been reported. However, the response of confined pure liquids to shear suggests solid-like behavior at sufficiently small gap separations. Shearing studies using the SFA2show that the shear stress depends strongly on the number of layers between the shearing surfaces. Measurements of shear viscosities of cyclohexane between mica surfaces show that deviations from the bulk shear response begin at film thicknesses below 5-10 particle diameters. Increased structuring is suggested when the walls are separated by three to four molecular layers, and a critical shear stress is required to initiate shearing. The value of the critical shear stress increases by an order of magnitude for each layer Difficulties in studying the equilibrium structure of confined fluids generally arise from the small particle and gap sizes required to observe structure. For many situations, colloidal suspensions offer several advantages over molecular systems in studying equilibrium structure. The typical colloidal particle size is on the order of a wavelength of visible light, making the particles readily observable with standard light scattering or optical microscopy techniques. The tendency of monodisperse (10)Courtemanche, D. J.;van Swol, F. Phys. Rev. Lett. 1992,69, 2078. (11)Henderson, J. R.;van Swol, F. Mol. Phys. 1984,51, 994. (12)Groot, R. D.; Faber, N. M.; van der Eerden, J. P. Mol. Phys. 1987,62,861. (13)van Megan, W.; Snook, I. J . Chem. Phys. 1976,66,813. (14)Lupkowski, M.; van Swol, F. J . Chem. Phys. 1990,93,737.
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colloidal systems to undergo phase transitions much like those observed for molecular systems is well established.16-18 In particular, colloidalsuspensions of uniform spheres a t large enough particle number density exhibit long-range order as evidenced by Bragg scattering and show rigidity as evidenced by the presence of a zerofrequency shear m o d u l u ~ . ~ ~ , ~ ~ Colloidal particles in the submicrometer size range are Brownian and sample an ensemble of states, making them amenable to treatment by statistical mechanics. BradyZ1 has recently shown that the osmotic pressure of a colloidal dispersion can be interpreted as the isotropic part of the macroscopic particle stress. Vrij et aLZ2have shown that structure and particle interactions in colloidaldispersions may be treated in the same manner as simple liquids and that the solvent properties appear as parameters in a potential of mean force between the particles. Thus, colloidal suspensions can serve as good model systems for investigating the equilibrium properties of molecular fluids. From a combination of computer simulation^,^^*^^ statistical mechanical prediction^,^^ and experiments with monodisperse model s y s t e m ~ , ~ ~the - ~ equilibrium ’,~~ behavior of many colloidal systems has been shown to be well characterized for both repulsive and attractive particle interaction pair potentials. Agreement between experiment and theory is generally good, with discrepancies apparently arising from the choice of the interaction potential used to model the system. The interaction of colloidal particles with a solid substrate, which is a problem that has relevance to diverse technologies from paper coating to ceramics processing in addition to being a good model system for molecular systems, has seen little investigation. In recent experimental studies on confined ~ o l l o i d s ,attention ~ ~ - ~ ~ has been given to the formation of thin colloidal crystals and the structures associated with them. Colloidal crystals have been produced by confining bulk crystalline suspensions within small-angled wedges of two flat glass plates. The morphology, orientation, and number of layers of a confined colloidal crystal were found to be strongly dependent on wall separation and bulk suspension properties. Surfaces appear to aid the formation of crystalline face-centered cubic (fcc) or body-centered cubic (bcc)structures. While providing some evidence for wallinduced structure in colloidal crystals, none of these studies has explored how walls alter the phase behavior of colloids. No studies have reported changes in phase behavior of confined suspensions through control of the thermodynamic state of the suspension in the bulk phase. (15) Chaikin, P. M.; Pincus, P. A. Liq. Cryst. Fluid 1984,4, 971. (16) Furusawa, K.; Tomotsu, N. J. Colloid Interface Sci. 1983,93, 504. (17) Kose, A.; Hachisu, S.J. Colloid Interface Sci. 1974, 46, 460. (18) Russel, W. B. Phase Transitions 1990,21, 127. (19)Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (20) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1989; Vol. I. (21) Brady, J. J. Chem. Phys. 1995, 98, 3335. (22) Vrij, A.; Nieuwenhuis, E. A.; Fijnaut, H. M.; Agterof, W. G. M. Discuss. Faraday SOC.1978, 65, 101. (23) Kremer, K.; Robbins, M. 0.;Grest, G. S. Phys. Rev. Lett. 1986, 57, 2694. (24)Vlachy, V.; Haymet, A. D. J. Chem. Phys. Lett. 1988,146, 32. (25) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. (26) Voegtli, L. P. Electrostatic,Electrokinetic, and Ordering Pmperties of Concentrated Suspensions of Polystyrene Latex; Ph.D. Thesis, 1989. (27)Pansu, B.;Pieranski, P.; Strzelecki,L.J . Phys. (Paris) 1983,44, 531. (28) Murray, C. A.;van Winkle,D. H.;We&, R. A. Phase Transitions 1990,21, 93. (29) Monovoukas, Y.; Gast, A. P. Phase Transitions 1990,21, 183. (30) Monovoukas, Y.; Gast, A. P. Langmuir 1991, 7, 460.
-2
Figure 1. Arrangement of the simulation cell showing twodimensional and three-dimensionalviews ofthe gap plates and pressure-driven walls.
Here, we report both computer and experimental studies of the phase behavior of charge-stabilized colloidal suspensions held between narrowly spaced surfaces at welldefined bulk conditions. The simulation technique employs a gap of fixed thickness placed within a suficiently large reservoir of bulk fluid such that the fluid in the gap is held at a fixed chemical potential facilitating comparisons with experimental systems of a similar configuration. The simulation uses molecular dynamics and is thus ideally suited for investigations of confined molecular fluids as encountered in recent investigations oftribology.2 We present predictions of the phase behavior of particles interacting with a screened Coulomb potential and compare the predictions to experiments of confined colloidalsuspensions where the solid and fluid phases are easily observed. Agreement between experimental and predicted behavior demonstrates the applicability of the simulation technique and provides new insight into the influence of surfaces on fluids. 2. Simulation Model The properties of a fluid confined between two surfaces in equilibrium with bulk fluid are determined from a threedimensional molecular dynamics simulation where the chemical potential of the fluid in the confining region is controlled by fixing the pressure and temperature of the bulk fluid. The simulation method implements the isothermal-isobaric ensemble ( N p nwhich has been used in the study of bulk fluid behavior, surface adsorption, and crystallization of fluids on solid surfaces.13 “ w o planar walls are placed parallel to one another at opposite ends ofthe simulation cell,bounding the system in one direction. The walls fluctuate, exerting a constant external pressure on the fluid between the walls. Periodic boundary conditions are implemented in the two directions parallel to the fluctuating walls. To study the thermodynamic behavior of particles confined to a gap of fixed separation while in contact with a bulk reservoir of suspension, two fured plates are placed between the two fluctuating walls that exert constant pressure, far enough away from the pressure-driven walls to avoid interactions. Particles move in and out of the gap, to and from the bulk reservoir. The arrangement of the pressure-driven walls and the fured gap in the simulation cell is shown in Figure 1.With a periodic boundary condition in the y-direction, the f s e d horizontal plates in fact introduce two gaps of equal or different size depending on the spacing. Particles are initially placed outside of the gap and allowed to flow into
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Langmuir, Vol. 11,No. 1, 1995 113
and out of the gap from the z-direction. Particle number (N),external pressure @), and temperature (2') are fixed
for the entire simulation cell. As a result, within the fmed gap, the volume, temperature, and chemical potential are constant. This technique is novel in its approach to directly control the chemical potential within the confining region with the bulk. The more intuitive solution of using a grand canonical ensemble to fm the chemical potential, as used successfully in many other studies involving low density phases, is not a good approach for the dense systems that we deal with here. The price one pays in our method to control the bulk state is that one is forced to simulate a rather large reservoir. However, this disadvantage is offset to some extent by the fact one can simultaneously study single wall phenomena at the fluctuating walls.'O In addition, the arrangement (Figure 1)allows one to study additional dynamic phenomena relating to the transport of particles into and out of the gap. The thermodynamic behavior of charge-stabilized colloidal suspensions at low ionic strength is governed by repulsive electrostatic forces between particles which are generally much greater than the Brownian forces imposed by solvent molecules or the attractive van der Waals f o r c e ~ . l ~ Solvent , ~ ~ ~ ~molecules, l due to their small size relative to the particles, act as a temperature bath for the colloidal particles under equilibrium conditions. As a result, colloidal systems can be modeled as a pseudo-onecomponent system in which the particles interact via a pairwise electrostatic potential. The interaction potential used here is the Yukawa potential, which stems from the solution of the linearized Poisson-Boltzmann equation for two charged spheres in an electrolyte interacting via Coulomb potential^.^^,^^ The Yukawa potential is given by
where r is the center-to-center particle separation, u is the particle diameter, and K is the inverse screening length of the electrostatic double layer. The characteristic energy UOis given by UO= Z E E O Ywhere O ~ UYO , is the particle surface potential and E and EO are the dielectric constant of the solvent and permittivity of vacuum, respectively. The screening length, 1/~, is expressed in terms of the electrolyte properties as
where ni is the number density of a screening ion of species i , and qi is the charge of a screening ion. An integrated form of the potential in eq 1 describes the interaction between a particle and a smooth repulsive wall, given bY4O
U W b )= uo,wexp[-Kb - a)]
(3)
where UO,, = U O Y J Y OY, , is the surface potential of the wall and y is the normal distance between the wall and the center of the particle. Equation 3 is also applied to particles interacting with the fluctuating walls in the z-direction. Interactions of particles with any wall occur such that the force experienced by the particles is always normal to the wall. A problem created by the arrangement (31)Vold,R. D.; Vold,M. J. Colloid andInterface Chemistry;AddisonWesley, Inc.: London, 1983. (32)Glendinning,A. B.;Russel,W. B. J . Colloid Interface Sci.1982,
93,95.
ofthe fmed gap plates is modeling the interaction between particles and the edges of the plates such that particles approaching and leaving the gap do not experience a discontinuous potential. To remedy this problem, the edges of the gap plates are modeled as particles, so that the particles experience a force directed away from the gap that is equivalent to a single particle-particle interaction. As a result, a particle sees a continuous potential as it moves in and out of the gap. In addition, the gap plates are opaque. Particles on one side of a gap plate do not interact with particles on the other side of the plate. Particle motion is governed by Newton's equations of motion given by
where m is the particle mass, ai is the particle acceleration, Fi is the force on the particle, and ri is the particle position. Newton's equations of motion were solved approximately using a leap-frog Verlet forward difference algorithm.33 Particle accelerations are obtained from the interaction potentials in eqs 1 and 3 according to
The fluctuating walls interact with the particles via the potential in eq 3 and move with acceleration a , and wall mass m , equal to the particle mass according to14
where Fed is the constant external force on the walls for the applied pressure and Fi,, is the force resulting from a particle-wall interaction. As a time-saving device, "cut-and-shifted" versions of the potentials are used in the simulations. All pair interaction potentials are cut at a separation, r, = 3u, where the value of the potential is less than 0.001 of its contact value. For r > r,, the potential is zero. For r 5 r,, the potential curve is shifted so that the value of the potential at the cutoff distance is zero; i.e. the cut and shifted interaction potential is U(r)- U(r,). The resulting discontinuity in the potential gradient is not expected to contribute significantly to errors in particle t r a j e ~ t o r i e s . ~ ~ 3. Experimental Procedure An experimental cell similar to the glass wedge used by was developed to confine and examine colloidal Pieranski et al.34 polystyrene suspensions within a thin gap while providing a bulk reservoir of suspension in contact with the confiningregion. The gap was formed from optically smoothglass slidesmeasuring 75 mm x 25 mm x 1 mm that were thoroughly precleaned in a base bath (2.5 w t % KOH) and neutralized in 10% HC1. One slide was cut to half its length and glued t o a full-length slide at one end. At the other end of the half-slide, a gap was formed betweenthe two glass slideswith two uniformglassmicrospheres of 20 pm in diameter positioned at the corners of the half-slide. This providedthe opening for suspension t o flowinto from a bulk reservoir of suspension. The bulk reservoir measuring 25 mm x 25 mm x 5 mm was constructed at the open end of the gap. Plates were sealed together with a nonleaching silicone adhesive, except at the open end in contact with the reservoir. The gap region measures 35 mm in length and 25 mm in width, with a maximum thickness of 20 pm and an angle of inclination of approximately 0.030" as set by the microspheres. The completed glass cell is diagrammed in Figure 2. Individual glass cellswere constructed for each suspension sample used. (33)Allen, M.P.; Tildesley, D. J. Computer Simulation ofliquids; Clarendon Press: Oxford, 1987. (34)Pieranski, P. Phys. Rev. Lett. 1980,45,569.
Hug et al.
114 Langmuir, Vol. 11, No. 1, 1995 glass plate\
area of enlargement
~ -reservoir --, microsphere
5u *rurpenrmn A & usp pension
0 0.7
0.9
0.8
B
1.0
$"I+:
Figure 2. Schematic of glass cell with wedge-shaped gap in contact with reservoir.
Colloidal suspensions were prepared from monodisperse polystyrene latices measuring 229 nm in diameter dispersed in aqueous media. The particles were synthesizedby the method given by Homola and James36 and then dialyzed against one of two aqueous M ionic strength solutions after cleaning: a M KC1 solution (dialysate pH = 6.7) and a carbonatebuffer M Na2C03 and 4.0 x M NaHC03 consisting of 2.0 x (dialysatepH = 9.5),yielding a value of u~= 25.0 as determined from eq 2. The volume fraction of each suspension was then raised until the particles were completely ordered. Suspensions M HCl (suspensionA) have of particles in equilibriumwith a bulk freezing volume fraction of 0.34. The particle electrophoreticmobilityis-0.57pmcm/(Vs)asdeterminedonaCoulter Delsa 440 automated electrophoresisinstrument. By use of the technique of O'Brien and White,36 this mobility corresponds to an estimated surface potential of -10 mV. Suspensions equilibrated against the carbonate buffer (suspensionB) have a bulk freezing volume fraction of 0.30. The electrophoretic mobility of these particles is -2.69 pm cd(V s) corresponding to a surface potentialof -50 mV. The measured values ofthe freezingvolume fractions are in good agreement with values predicted by perturbation methods.26 Suspensions used in the experiments of phase behavior in confined spaces were prepared at various volume fractions from the ordered suspensionsby diluting with the dialysate. Empty cells were illuminated with monochromatic light and the gap thickness was determined by countingfringes produced by the constructive interferenceof light created by the mismatch in optical densities of glass and air. The interference condition for maxima in a glass/air/glass configuration is given by37
(7) where d is the gap thickness, Aii, is the wavelength of light in air, nglasSis the index of refraction of glass (nglass = 1.51, and m is the fringenumber. Fringe patterns were obtained using light with a wavelength of 525 nm. Suspensions of known volume fractions were loaded into the reservoirs of the individual glass cells and allowedto flow withinthe gap. Particlevolumefractions of 0.26, 0.28, 0.30, and 0.32 were used for suspension A (bulk freezing volume fractionw = 0.34)and volume fractionsof 0.22, 0.24,0.26,and 0.28 were used for suspension B (w= 0.30).All suspensionswere fluidsat volume fractionsbelow their respective bulk freezing volume fractions. Each sample was allowed to equilibrate for 1 week. The location of the boundary between ordered (iridescent)regions and disordered (opaquewhite)regions and the corresponding fringe number of the boundary was determined for each sample. The gap thickness at the orderdisorder boundary was calculated from eq 7. An error in the measured gap thickness off approximately 350 nm results from an uncertainty in the fringe count of two fringes. 4. Experimental Results The maximum gap thickness at which a suspension crystallizes was determined as a function of bulk volume (35)Homola, A.;James, R. 0.J.Colloid Interface Sci. 1977,59,123. White, L. R. J. Chem. Soc., Faraday Trans. 2 (36)O'Brien, R.W.; 1978,74, 1607. (37)Halliday, D.;Resnick, R. Physics Part 2, Wiley and Sons: New York, 1978.
Figure 3. Maximum gap thickness of the solid-liquid phase boundary as a function of bulk volume fraction for suspension A (Yc= -10 mV, @' = 0.34)and suspension B (YO= -50 mV, &- = 0.34). The gap thicknessis expressed as number ofparticle diameters, u, and the bulk volume fraction, @,is scaled t o the appropriate bulk freezing volume fraction, @'. fraction for the two colloidal systems of different surface potential. The freezing points are plotted in Figure 3, where the maximum gap thickness of the crystalline region, he is scaled to the particle diameter u and the corresponding bulk volume fraction, p , is scaled to the bulk freezing volume fraction &". For a gap thickness h > hf, the suspension is fluid. For h hf,the suspension iridesces, indicating crystalline structure. Both colloidal systems A and B have substantially different surface potentials yet demonstrate similar qualitative behavior in that there is a dramatic lowering of the freezing volume fraction as the thickness of the confining region is decreased. 5. Simulation Procedure 5.1. Bulk Simulations. Preliminary constant pressure simulations were runwithout fxed gaps to determine the bulk freezing pressure. The system simulated was suspension B with UK = 25 and temperature T* = kTIU0 = 0.0025. Initially a fluid of 640 particles was placed between the two fluctuating walls of dimensions L, = 50 and Ly = 100 separated by a distance L, = 25u. Periodic boundary conditions apply in the x and y directions. Random velocities were assigned to each particle based on a Maxwellian distribution. The system was first equilibrated at a fixed pressure ofp* =pd/Uo = 0.01 over 100 000 steps using a reduced timestep At = 0.005t, where t = (ma2/U,J-1'2. At each timestep, the velocities were scaled to the desired temperature using the constraint method of Brown and Clarke.38 The final configuration for the initial run at p* = 0.01 served as the starting point for a series of subsequent simulation runs performed in increments of Ap* = 0.001. The final configuration at each pressure served as the initial configuration for the run at the next higher pressure. Equilibration was reached within 50000 At. The progression of pressures continued into the crystalline phase. The bulk freezing pressure was found to be approximately p* = 0.024 as determined by particle configurations showing structure throughout the system. At this pressure, the equilibrium volume fraction is 0.294 as determined by averaging the volume of the simulation cell every 25 timesteps over the final 5000 At. 5.2. Simulations of Confined Particles. Simulations of particles confined within fixed gaps of varying = thickness were run with gap walls of dimensions Lx,gap 50 and Lz,gap= 52/30. The walls are periodic in the x-direction. The dimensions of the gap were chosen to accommodate the (111)planes of a fcc crystal. However, we expect that other planes, such as (100) or (110) could also be fitted without much distortion, since there is no (38)Brown, D.;Clarke, J. H. R. J. Chem. Phys. 1984,51, 1243.
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periodic boundary condition in the z-direction. A system of 1280 particles was found to provide a n acceptable balance between obtaining bulk behavior near the gap and economizing computation time. Simulations with the fxed gap were run at UK = 25 and temperature T* = 0.0025, corresponding to suspension B, at reduced pressures ranging fromp" = 0.01 (bulk fluid) to the bulk freezing pressure p* = 0.024 (as determined by bulk simulations) progressing in increments of Ap* = 0.001 along the freezing path. Wall separations between h = 5u and h = 100 were investigated in increments of Ah = 0 . 2 5 ~and held constant over the entire range of pressures. Runs beginning a t p * = 0.01 were performed over 100000 At starting from a random initial configuration. Subsequent runs at higher pressures were performed at 50000 At beginning with the final configuration of the previous run. Density profiles normal to the gap plates, were calculated for particles within the gap according to
6 5 4 a
2 0
6
(8) 5
where (N(y))is the average number of particles between the gap walls of area A,,, in a slice of thickness Ay located between y - Ay/2 and y Ayl2. Averaging took place every 25 timesteps using a slice of Ay = h180. Two-dimensional order parameters were applied to individual layers within a particle configuration to help determine if crystal or fluid exists. The order parameters OP is given by:
+
(9)
4
P
2 0
E 4 0. 2
wherex andzj are the center-to-center distances of particle j with respect to some origin and h and 1 are the reciprocal lattice parameters in the x - and z-direction, respectively,
0
-2.875
0.0 YIO
2.875
and Ngap is the number of particles within the layer. The maximum possible value of the order parameter is 1for a perfectly crystalline hexagonally closed-packed layer. The reciprocal lattice parameters h and 1 are determined for a single layer by projecting the spheres as twodimensional disks onto a single plane and calculating according to
h
1-& l h
(10)
where the value of 0.9068is the maximum packing fraction for 2-D disks. The reciprocal lattice parameters h and I correspond to hexagonally closed-packed (111)planes of particles. Order parameters are calculated for instantaneous particle configurations at equilibrium. Timeaveraging of order parameter calculations over many configurations during a single simulation run was performed. I t was found that at equilibrium, values of the order parameters varied less than 2% from one instantaneous configuration to another. Equilibration was determined when no significant changes in the density profiles, order parameters, and boxvolume were seen over at least 20 000 timesteps. Runs of 50000 timesteps were sufficiently long to reach equilibrium with the small increments in pressures that were used. To show that the final equilibrium state was not path dependent, several runs were performed where the gap thickness was increased or decreased systematically at a fxed bulk pressure. Although the results are not presented here, there were no significant differences in density profiles and order parameters between equili-
Figure 4. Gap density profiles (see eq 8) atp* = 0.016: (a)h = 5 . 0 ~(b) ; h = 5.250;(c) h = 5.50; (d) h = 5.750; (e) h = 6.0~; (0 h = 6.25~."he origin is located at the center of the gap.
brated systems generated either by varying the pressure a t constant gap thickness or by varying the gap thickness a t constant pressure. 6. Simulation Results The gap density profiles, &f), show sharp oscillations at the gap thicknesses investigated, indicating that the particles form layers parallel to the confining plates (see Figure 4). Particle configurations showing the structure
116 Langmuir, Vol. 11, No. 1, 1995
Hug et al.
AX
t" Figure 5. In-plane configuration of particles in gap of thickness h = 5.00 atp* = 0.016: (a)first layer, closest to a gap wall; (b)
second layer from wall.
Figure 6. In-plane configuration of particles in gap of thickness h = 5.50 atp* = 0.016: (a)first layer, closest to a gap wall; (b)
second layer from wall.
of the individual layers are used to determine regions of solid or fluid between the two fxed plates. Particles within a gap are considered to be completely crystallized when each layer of particles is crystalline. There is a strong interplay between the gap size and ability to form crystalline layers within the gap. The phase behavior of the confined particles is reflected in the density profiles. Figure 4 contains plots of reduced density (eq 8 ) versus position in the gap illustrating how the density profile changes with increasing gap thickness at a fxed bulk pressure. All profiles are shown for a fxed pressure ofp* = 0.016 (#m = 0.260) which is well below the bulk crystallization pressure ofp* =0.024 (#fm =0.294). Indeed, for all gap thicknesses atp* = 0.016, the bulk outside of the gap remains fluid. Distinct peaks in the density profile occur at gap thicknesses corresponding to an integral number of layers, as seen in Figure 4a for a gap thickness of h = 5.00. At this gap thickness, crystal spans the gap. The plots in Figure 5 show the in-plane (x-z) structure of the first and second layers in a gap of thickness h = 5.00. At larger separations of h = 5.250 (Figure 4b) and h = 5.50 (Figure 4c), the density peaks broaden and decrease in magnitude and the layers lose their structure. The lack of ordering in the first and second layers at a gap thickness ofh = 5.50is shown in Figure 6. Alayer nearest to a gap wall has some ordering, but the second layer is disordered. Increasing the gap thickness to h = 5.750
Figure 7. In-plane configuration of particles in gap of thickness h = 6.00atp* = 0.016: (a)first layer, closest to a gap wall; (b)
third layer from wall (middle layer).
allows for an additional layer to form (Figure 4d). However, at h = 5.750, the gap is still apparently not wide enough for complete crystallization to take place. At h = 6.00 (density profile in Figure 4e), a crystal of five layers has formed. To illustrate this, Figure 7 shows two crystallized layers, one at a gap wall and one at the center of the gap. The configuration plot in Figure 8 shows the entire equilibrated system at h = 6.00, with layers of particles in the gap in contact with bulk fluid outside of the gap. Beyond h = 6.00 at this particular bulk pressure, the center of the gap becomes fluid and increasingly bulklike as the wall separation is further increased. The density profile for h = 6.250 is given in Figure 4f. Higher pressures are required to form crystals with a greater number of layers. Two-dimensional order parameter values for five-layer equilibrium configurations generated atp* = 0.016 and gap thicknesses ofh = 5.50,5.750,6.00, 6.250, and 6.50are given in Table 1. The tabulated values are layer averages for a single configurationat equilibrium. The highest degree of order is seen in the configuration generated at h = 6.00, in agreement with the density profiles. A detailed phase diagram covering a large range of bulk pressures (p* = 0.010-0.024 with steps of 0.001) and gap thicknesses (h = 5.0-10.00 with steps of 0.250) is given in Figure 9. Lines have been drawn to indicate regions of different numbers of particle layers. The points correspond to the minimum pressure at which crystal is seen for a particular gap thickness. At pressures beyond these points, the particles in the gap are completely crystallized. In the phase diagram, the shaded regions are state points where the particles in the gap have a full crystalline structure. Fluid is seen in the unshaded regions. The solid and fluid regions are not clearly defined close to the boundaries between two integral number of layers or beyond the bulk crystallization point. Particles crystallizein layers of (111)planes parallel to the confining walls. There appears to be no obvious pattern of the stacking of (111)planes relative to one another, suggesting that the (111)planes are randomly registered. Kremer et al.23and Clark et al.39see randomly registered stacks of (111)planes in bulk colloidal crystals. At pressures far beyond the bulk crystallization point, crystal orientations other than stacks of (111)planes were observed in gaps of thicknesses that do not accommodate an integral (39) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979,281,57.
Langmuir, Vol. 11,No. 1, 1995 117
Freezing of Colloidal Suspensions
bulk fluid
gap walls
Figure 8. Configuration of bulk and confined particles, h = 6.00, p* = 0.016. Table 1. Order Parameters of Layers of Confined Particles at Different Gap Thicknesses,p* = 0.016 h (a> layer 1 layer2 5.50 5.75 6.00 6.25 6.50
0.560 0.625 0.918 0.886 0.905
a Average
0.775 0.520 0.839 0.868 0.782
layer3
layer4
layer5
(OP)"
0.610 0.664 0.862 0.872 0.746
0.698 0.523 0.941 0.820 0.777
0.711 0.687 0.893 0.889 0.904
0.671 0.604 0.891 0.867 0.823
bulk volume fraction 10'
0.240.25 0.26 0.27 : : , : !
18 layers
0.010
l I
l I
1 I
=t ;1 .{ ' ==t=
!
%
did
5 -
cxpcrinicnt
.
~iinulniiun
1
1
I
I
I
I
m !(
0.28 I
I I
liquid
l o --
0
value over all layers.
, I
15
:
0.29 w
I
0.015
P*
0.020
0.025
Figure 9. Phase diagram generated from simulations of confined particles at bulk pressure,p*, (equivalent bulk volume fraction, 4") and gap thickness, h. Shaded regions correspond to solid within the gap. Unshaded regions correspond to fluid within the gap. The bulk freezing volume fraction is 0.294.
number of layers of (111)planes. This is consistent with structures observed by Pieranski et al.34and Murray et a1.28
7. Discussion and Conclusions We have demonstrated by simulation and experiment the phenomenon of capillary freezing of colloidal particles at volume fractions significantly lower than the bulk freezing volume fraction. A comparison of the experimental data for suspension B with the simulation data is shown by a plot of hp/a versus @/&- in Figure 10. The experimental data are duplicated from Figure 3. The simulation data have been generated from the phase diagram in Figure 9 and correspond to the minimum volume fraction required to crystallizea particular number of layers. Bulk pressures have been converted to the appropriate bulk volume fractions. Solid and fluid regions are indicated on the graph. Quantitative differences between simulation and experiment can be attributed to uncertainties in measuring gap thicknesses in the experiments and in determining
freezing volume fractions, both in experiment and in simulation. Differences could also in principle be due to the values chosen for the wall and particle surface potential. For a repulsive colloidal system, a short-ranged wall surface potential determines the effective gap thickness seen by the particles. A system with an effective gap thickness at one surface potential is expected to exhibit the same phase behavior as another system with the same effectivegap thickness at a different wall surface potential. The chosen wall potential for the simulations was -50 mV. The wall surface potential, V,I was later estimated from work done by Scales et aL40on streaming potentials of glass in electrolytes. The estimated value of Y, for a solution a t M ionic strength and pH = 9.5 is about -90 mV. Work done by Prieve et al.41and Schumacher et al.42shows that the average distance between a charged particle and a repulsive wall is a weak function of the strength of interaction between wall and particle. Therefore, a different choice ofwall potential for the simulations is not expected to significantly alter the phase behavior. Moreover, due to the large value of OK and the steepness of the wall-particle potential, changes in the effective gap thickness are expected to be small for the system studied. Colloidal crystallization by confinement has been demonstrated in suspensions as low as 70% of the bulk freezing volume fraction. The lower limit in volume fraction where one can see crystallization between two surfaces was not explored but is believed to be the (40) Scales, P. J.; Grieser, F.; Healy, T. W. Langmuir 1992,8,965. (41) Prieve, D. C.; Bike, S. G.; Frej, N. A. Faraday Discuss. Chem. SOC. 1990, 90,1. (42) Schumacher, G. A.; van de Ven, T. G. M. Langmuir 1991, 7, 2028.
118 Langmuir, Vol.11,No.1, 1995 minimum volume fraction required to crystallize a monolayer. The minimum gap thickness required to form a 2-D crystal depends on the wall-particle interaction which determines the wall-particle separation and effectivegap thickness. The presented simulation method is extremely useful to study confined fluids consisting of particles with other interaction potentials in that the thermodynamics of the system are controlled by a reservoir of bulk fluid. Reservoirs have been used in other simulation techniques, but not with a true control of the bulk properties. These methods rely on insertions and deletions of particles within the confining region to control the chemical potential.33 In our technique, the chemical potential within the confining region is fxed by a constant external pressure applied to the entire system and maintaining direct contact of the confined particles with the bulk reservoir. The flow of particles in and out of the confining region is dictated by the equations of motion rather than stochastic means. Direct comparisons can be made between confined phase behavior and bulk phase behavior within a single simulation run. The results presented have implications beyond suspensions and have a direct connection to work on molecular
Hug et al. systems. The colloidalsystems studied here have density profile oscillations with a periodicity of one particle diameter. In force-distance measurements on liquid films in the surface forces apparatus, density oscillations are manifested as oscillations in the force required to hold plates at a given separation. A transition to a solid occurs in our studies as the wall separation approaches a value corresponding to an intergral number of layers. This transition will be accompanied by large changes in the flow properties. Shear studies of molecular fluids by van Alsten and Granick2 show order of magnitude increases in the apparent viscosity of the confined fluid at gap spacings corresponding to an integral number of layers. However, due to the current lack of structural information, one cannot conclusivelystate that structural changes take place. Small changes in film thickness to a nonintegral number of layers restore the liquid state.
Acknowledgment. This work was supported by the
U.S.Department of Energy through the Materials Research Laboratory at the University of Illinois, UrbanaChampaign, under Grant Number DEFG 02-91 ER45439. LA9305132
