J. Phys. Chem. 1991, 95,4819-4826
4819
A Monte Carlo Simulation Study of the Interaction between Charged Colloids Carrying Adsorbed Polyelectrolytes M. K. Granfeldt,* Bo Jonsson, and C. E. Woodward Physical Chemistry 2, Chemical Center, Box 124, 221 00 Lund, Sweden (Received: September 7, 1990)
We have used Monte Carlo simulations to study the interaction of two uniformly charged colloidal particles with adsorbed oppositely charged polyelectrolytes. The latter are modeled as point charges connected via a harmonic potential. A predominantly attractive net force is found. The force separates into an attractive term due to bridging and electrostatic correlations and a repulsive term of direct entropic origin. The relative importance of these interactions is investigated as a function of Separation, addition of simple electrolyte, changes in polyelectrolyte structure, and colloid charge. At large separations the force is found to be determined by electrostatic interactions, but at shorter separations the bridging by polyelectrolytes stretching between the colloids is the dominant contribution. This bridging attraction is large and significant in comparison with the ordinary van der Waals interaction.
Introduction Polyelectrolytes have become very useful as modulators of colloidal interactions and the interaction of large “particles- in general. One example is the stabilization of colloidal solutions by added polyelectrolyte. This is usually accomplished by electrostatic and steric interactions. The magnitude of the resulting colloidal interaction depends on a number of factors, e.g., polyelectrolyte charge and flexibility and ionic strength. Polyelectrolytes may also act as potent flocculants as in the pulp industry and in sewage stations; the range of applications is indeed wide. These qualities are no doubt also important in biological systems where many polyelectrolytes occur. Examples of such natural polyelectrolytes are the polynucleotides, RNA and DNA, and polysaccharides. The latter often appear as the hydrophilic part of glycolipids found on the surface of cells. Thus, our understanding of biological processes such as protein-DNA and cell-cell interactions, as well as of processes in many technical applications, requires a knowledge of the forces involved. In most cases the usage of polyelectrolytes is based on empirical observations rather than the prediction of theoretical models, which is understandable in view of the complexity of these systems. Compared to ordinary electric double layer interactions in the presence of monovalent atomic or small molecular ions, the polyelectrolyteadds the extra complications of connectivity and flexibility, which have largely escaped a thorough theoretical treatment. Visual studies of both coagulation and flocculation of colloidal suspensions with polyelectrolytes have quite an old history.’ More recently, small-angle neutron scattering has been used to examine the organization of silica spheres, flocculated by polyelectrolyte.2 Direct measurements of the force between mica surfaces in the presence of polypeptides’ or in a micellar and polyelectrolyte solution4 have also been made. A slightly different system containing mixed ionic/nonionic surfactant micelles and strong polyelectrolytes has been studied by turbidimetric titration.5 Recently a simple model, suitable for theoretical studies of polyelectrolyte behavior, was introduced by Akesson et aL6 In this model a flexible polyelectrolyte is treated as a chain of point charges connected via a harmonic potential. The simplicity of the model makes it amenable not only to numerical simulations but also to an analytical mean field approach. The model was used to study the interaction between charged surfaces in the presence of oppositely charged polyelectrolyte. Results were (1) Pugh, T. L.; Heller, W. J . Polym. Sci. 1960, XLVII, 219.
(2) Cabme, B.; Wong, K.; Wang. T. K.; Lafuma. F.; Duplessix, R. Colloid Polym. Scl. 1988, 266, 101. (3) Afshar-rad, T.; Bailey, A. 1.; Luckham, P. F. Colloids Surf. 1988, 31, 147. (4) Marra, J.; Hair, M. L. J . Phys. Chcm. 1988, 92, 6044.
(5) Dubin, P. I.; The, S.S.; McQuigg, D. W.; Chew, C. H.; Gan, L. M. Lungmyir 1989, 5 , 89. (6) Akcsson, T.;Woodward, C. E.; Jansson, B. J . Chcm. Phys. 1989,91, 2461.
obtained from Monte Carlo simulations as well as from an extended Poisson-Boltzmann (PB) equation which took the polyelectrolyte connectivity into account. The latter property proved to have significant physical consequences. It was found from both simulations and analytical calculations that connectivity gave rise to a strong attractive contribution to the total force. This attraction was due to polyelectrolyte chains stretching between the charged surfaces which we refer to as bridging. It can be explained as a balance between entropic and energetic considerations.6 These theoretical findings are particularly interesting, as experiments have indicated a strong attraction between similarly charged particles in a solution of oppositely charged polyelectrolyte.* Though in some cases the attraction could be attributable to van der Waals forces, uncertainty in the choice of the Hamaker constant leads to ambiguities in this explanation, for example, in the case of negatively charged mica surfaces confining positively charged polylysine.’ In other cases, such as with myelin basic protein, the attraction was too large to be explained by van der Waals forces and the bridging mechanism described above may be a possible explanation. However, the experimental conditions are rather complex and it is extremely difficult to determine the relative importance of various contributions to the total force. The extended PB theory and the Monte Carlo (MC) simulations have also been applied to a system of planar charged surfaces onto which one end of the polyelectrolyte molecules is grafted.’ In these calculations the various contributions to the force between the surfaces were determined separately and the bridging contribution was found to be significantly larger than the van der Waals interaction. In the present study we use MC simulations to investigate the interaction between two spherical colloidal particles with grafted polyelectrolytes in a system where the polyelectrolyte and the particles are of comparable size. The model is kept simple with charged hard spheres to model the colloidal particles and where the chains consist of point charges connected via a harmonic potential. An explicit partitioning of the various contributions to the force is made and we also investigate the effects of adding simple electrolyte, changing the colloid charge, and the polyelectrolyte flexibility. The type of force calculations attempted here often suffer from numerical problems and two different ways of calculating the forces have been tested and compared. In the next section the model system is presented in more detail.
Model System and Simulation Details Figure 1 shows a simple linear polyelectrolyte, a poly(a1kanimine). It is not difficult to imagine more complex structures with extensive side chains or branching polymers, but a rigorous theoretical study of even this simple polyelectrolyte is probably un(7) Miklavic, S. J.; Woodward, C. E.; Jhsson, B.; Akesson, T. Macromolecules 1990, 23, 4 149.
0022-365419112095-4819302.50/0 0 1991 American Chemical Society
Granfeldt et al.
4820 The Journal of Physical Chemistry,Vol. 95, No. 12, 1991
Figure 1. Schematic picture showing how a real polyelectrolyte, in this case a poly(a1kanimine) is treated in the model. The hydrocarbon chain with its charged nitrogens is replacad by a harmonic potential acting
between point charges.
IO
e5 n
0
2
6 8 10 Distance (A) Figure 2. Probability distribution for the carboxylicoxygens in glutamic acid obtained from a molecular dynamics simulation of a small peptide in aqueous solution.* 4
tenable. In our model we essentially replace the complicated internal structure of the alkane chain with an effective potential that acts between the charged nitrogen atoms. This can be described in the following way. Let N be the number of charged groups on a chain of NtQ(atoms, such that each charged atom is separated by M neutral atoms. Consider such a section of uncharged atoms; if M is sufficiently large, then according to the central limit theorem the distribution of the end points becomes Gaussian. Thus the effective bond between charges can be described with a harmonic potential of the form K?. The force constant K describes the strength of the bond and thus the overall flexibility of the chain. It will enter as a parameter in our model and, in principle, can be adjusted to correspond to a certain charge per unit length. If the number of intervening atoms is small or if there are any steric restraints, an alternative form for the bonding potential, such as K(r - re)2is probably a better description. One way to obtain a bonding potential, specific to a particular polyelectrolyte, would be to extract the charge-charge distribution function from a full molecular simulation of the appropriate section of the polyelectrolyte chain. From such a digtribution function an effective bonding potential could then be inferred. Figure 2 shows the distribution between a pair of charges on a short polypeptide taken from a molecular dynamics simulation.* Its functional form would suggest a bond function of the type k(r - re)2. In future work we will pursue such a line of attack to determine bonding potentials. Many features such as hydrophobic interactions, hydrogen bonds, and intemal structural constraints due to trans and gauche conformations could be included in such an approach. For the present, however, we will concem ourselves with the qualitative behavior associated with connectivity and flexibility; thus, we maintain a simple Gaussian description. This has the additional advantage of leaving us with only a single adjustable parameter! (a) Tbe Model. In our simplified model, a polyelectrolyte chain is formed by joining N monovalently charged point particles with (8) Ullner, M.; Teleman, 0. To be published.
Figure 3. Snapshots taken from Monte Carlo simulations at 8,14, and 24 A separation, respectively. The colloidal particles are shown as two large spheres located at the Cartesiancoordinates(O,O,O) and (0,O.b) onto which chains of connected charges have becn attached. The monomers are drawn as small spheres but are treated in the calculation as point charges.
harmonic bonds. One end monomer of each chain is grafted to the surface of a colloidal particle with a similar type of bond, where the colloidal particle is modeled as namely, K(r an oppositely charged hard sphere with radius Rodl. Six chains are attached to each colloid and in each chain, N = 10. The two colloids are located at the Cartesian mrdinates (O,O,O) and (O,O&), respectively. The model system is depicted in Figure 3 with snapshots taken from MC simulations at three different separations. The particles and attached polyelectrolyte molecules are immersed in a solvent, which is treated as a dielectric continuum of relative permittivity, e,. The interiors of the colloidal particles are assumed to have the same dielectric constant as the surrounding medium; i.e., no image effects are included in the model. In all the calculations reported below, the value of the dielectric permittivity, c, was 78.3, modeling water at 298 K. The total charge on the polyelectrolyte, attached to each colloid, was fixed at 60 while the bare colloid charge was either -40, -60, or -80. In the major part of the calculations the colloid radius was set to 18 A but some simulations were performed with a radius of 60 A.
The potential acting between neighboring monomers in the chains is thus u w ( r ) = uhum(r)+ u&) = K$ + $/41~e,r (1) In the following we use the variable
rdn = ( $ / 8 1 ~ c , K ) ~ / ~
(2)
which is the separation at which Vw is minimum, to characterize
Interaction between Colloids and Carrying Polyelectrolytes
-
-c. . A
(c) Monte Carlo Simulations. The Monte Carlo calculations were performed in the canonical ensemble by using the Metropolis algorithm.I1 All monomers were allowed to move freely in three dimensions, whereas the point of attachment of each chain was constrained to sample the spherical surface of the colloid. An equilibration run, starting from a random distribution of monomers, consisted of at least 30 000 configurations/particle. The system was then simulated for an additional 70000-150 000 codigurations/particle during which the force was evaluated every 2000 configurations. For the full system a total simulation required approximately 40 min of CPU time on an IBM 3090 with vector facility.
Al\ 4
8 Distance (A) F'i4. Different components of the monomer-monomer potential, cf. eq I , as a function of separation. The different curves are uhm ud(---), uW (-) and the probability distribution P(r) = $ e x p (-e-),
(-bmd/kBT)
the bond strength, rather than K. Figure 4 shows the harmonic and electrostatic components of the nearest-neighbor pair interaction, together with their sum and the probability distribution P(r) = 9 exp(-,9ubd). The nearest-neighbor potential has a minimum at r = r& = 5 A in Figure 4, but the maximum in P(r) appears at approximately 7 A due to the weighting factor 9. One peculiar property of the model is that the uncharged parts of the polymer, the "bonds", are able to pass through each other and also penetrate into the colloidal particle. The latter is more likely whenever a pair of connected monomers approach the surface, which happens frequently, and increases with rmjn,On the other hand, it is not unreasonable to assume that some penetration of hydrocarbon chains into the interior of the aggregates will occur in reality. (b) Adding Electrolyte. In most experimental situations the solution also contains other simple ions. Extending the present model system to include free salt particles creates a few technical problems. First, when oppositely charged ionic species are present, it is no longer possible to treat the monomers in the chain as point charges. One has to assign hard-core radii to both monomers and salt particles, which increases the number of parameters. Furthermore, the computing demands increase with the number of particles. In addition, a simulation with salt would have to be performed by enclosing the system in a box with either periodic or hard wall boundaries. To avoid artifacts from whatever boundaries are chosen, the box has to be large and even at low salt concentrations the required number of particles would lead to a large increase in computation time. This problem would be greater with larger separation between the colloidal particles. The alternative approach, used here, is to model the screening effect of the salt by using an effective charge-charge potential. We have chosen a screened Coulomb potentialg u,j
The Journal of Physical Chemistry, Vol. 95, NO. 12, 1991 4821
= expMR, + R,)I/ [(I + &)(I
+ K R ~ ) I ( Z , Z ~ * / ~exp(-Kr)/r . R ~ O ~ ~ )(3)
where 2, and Zj are the charges and R, and Rj their effective radii. In order to simplify our analyses we treat the monomers as point particles, which is still possible in the screened Coulomb description. K is the inverse Debye screening length, related to the salt concentration, cs, via K
= e(2NAcs/coc,kBT)'/2
(4)
The validity of the screened Coulomb approximation has not been tested explicitly for this model system, but for a single polyelectrolyte chain in a monovalent salt solution, it is an excellent approximation as shown in recent MC simulations.10 (9) Verwey, E. J . W.; Overbeck, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (10) Woodward, C. E.; JBnsson. Bo. Submitted Chem. Phys.
Force Calculations
Due to the difficulties in obtaining accurate values for the force from the Monte Carlo simulations at short separations between the colloidal particles, we investigated several different methods to calculate this quantity. We present two different methods below. For larger separations, where chains on different colloidal particles essentially did not overlap, we employed yet another approach. (a) Method I. Direct Derivative of the Free Energy. The Helmholtz free energy for the system is A = -kBT In ZAB (5) where ZAB= $...$exp(-Utol(6;RA,R~)/kBT) dRA dRB
(6)
This is the configurational integral at a separation of b. RA and RBcollectively denote the polyelectrolyte coordinates attached to the two colloids, labeled A and B, respectively. We partition the total potential into the following components Utot(hRA&) = UA(RA) + UdRB) + UAB(b;RA,RB) (7) Here (y = A, B) contains all interactions within system y. i.e., all monomer-monomer and monomer-colloid interactions for polyelectrolyte molecules attached to a single colloid. U, contains all the cross interactions between systems A and B and it is the only term in the Hamiltonian that depends on the separation 6. To obtain the force acting between the two subsystem we need to evaluate the derivative F = -dA/ab = kBT a In Z ~ , / a b
-$...SaUlol/d6 * exp(-UI,,/kBT)
dRA dRB/ZAB =
- ( a U ~ ~ / a b (8) ) This expression corresponds to the most direct derivation of the force. The average is straightforward to evaluate as long as the interparticle potentials are continuous. However, if one has a hard-core interaction between monomers and the colloid particles, an impulsive contribution arises involving a surface integral of the monomer-colloid contact density times a geometric factor. In order to avoid this awkward evaluation we used a softer, repulsive r-12 potential for the interactions between a given colloid particle and monomers attached to the other particle. However, monomers still interacted with the colloid they were attached to via a hard-sphere potential. Thus the modified monomerccolloid interaction reads 4n-C(r) =
Zcolle2 exPl-4r - Rcol,)l/[(1 + KRc011)4.R~0~~1 + ff/rI2 ( 9 ) We shall call the system employing this type of potential model I. The coefficient a was chosen so that the repulsive term was equal to 10kBTat a separation of &.The force will then depend on a,particularly so at short separations. (b) Method 11. The following force expression provides a simpler treatment of hard-core interactions and we applied it to ( 1 1) Metropolis, N. A.; Rosenbluth, A. W.; Rosenbluth, hi.N.; Tellm. A.; Teller, E. J. Chem. Phys. 1953, 21, 1087.
Granfeldt et al.
4022 The Journal of Physical Chemistry, Vol. 95, No. 12, 1991
.
‘pI
a
C
Figure 5. Schematic picture of an imaginary two-step process of increasing the separation between the spheres. (a) The initial separation with a fictitious wall, across which some monomers stmht. (b) The two subsystems arc separated a distance 66, causing changes in the electrostatic interaction between charges on either side and a stretching of the polyelectrolyte bonds. (c) The walls at the center are moved a distance b6/2 so that they coincide again.
tbemodelwhalltheshort-ran~~loidinteractions are of the simple hard-sphere type; we call this model 11. Furthermore, this method naturally partitions the total force into separate components, each with a physically well-defined source. For these reasons this method (and model) was the main one employed in this work. Method I was used mainly for comparison and checking, especially at larger separations, where the effect of the differences in the potentials would be unimportant. The midplane between the two colloidal particles divides the systems into two halves, which we label 1 and 2 (Figure Sa). When the distance between the sphms is to be changed by a small amount, 66, the process can be thought to take place in two steps. First, the two subsystems, 1 and 2, are separated by an infinitesimal distance 66 so that a gap is created between them; see Figure Sb. This process increases the separation between the colloids yet affects only the interactions acting acTo6s the midplane. We denote this interaction as WI2. The process requires a force
F = -(auI2/ab) (10) This force will contain two terms each of different origin. One is from electmtatic interactions acto6s the midplane and the other arises from bonds which stretch across the midplane; see Figure Sb (note that there is no hard-core contribution as the monomers are point particles). These two terms, which we denote by Fd and F-, respectively, will be calculated individually. In the second step the gap is closed by enlarging one or both of the subsystems while keeping the separation between the colloidal particles constant; see Figure Sc. This process only affects the integration limits of the configuration integral and the contribution to the force is proportional to the monomer number per unit length at the midplane, N ( 6 / 2 ) . Due to its origin (a change in available volume) we call this term the entropic contribution, F,* Thus, the total force has three distinguishable contributions F = -(aUlz/a6) + k~TN(6/2)= FeI + Fau, + FMt (1 1) When the two colloidal particles are at a finite separation, the average charge distribution around each is nonspherical. If one were to calculate the electrostatic interaction between the two halves using their average charge distribution, i.e., using a mean-field description, one would find a repulsive contribution. However, particles correlate with each other in such a way that low-energy interactions are predominantly sampled, which leads to a much less repulsive or even attractive force between the two halves. The harmonic contribution, Fhm, due to springs between connected monomers on the opposite side of the midplane, is always attractive for the harmonic potential Kg. The springs can
be pictured as pulling the two halves of the system together. The magnitude of this contribution is determined by two factors: the number of springs crossing the midplane and the amount of stretching in each one. The mechanism can be explained in the following way. With only one colloidal particle present or with two colloids at large separation, the chains would be collapsed onto the surface, due to the strong monomer-colloid attraction. Such a configuration has a very low entropy, but the cost in energy for sampling configurations off the surface is also large. With a second colloidal particle present, the chains have the opportunity to sample the region in the attractive well at the other colloid. This decreases the energy cost of a substantial entropy gain. The extent of bridging is of course affected by the separation. The most favorable case is found when the separation is of the order of the average monomer-monomer separation. When the separation increases, the springs have to stretch or an intermediate monomer must sit in an energetically unfavorable position away from both surfaces. With increasing separation the cost in energy increases and the bridging contribution decreases rapidly. (c) Cakulathgtbe Forceat LnrgeSeperatiOrra A Perturbth Approach. At larger separations where there is essentially no overlap between chains on different colloids, we chose an alternative means for evaluating the average in eq 8. This method was recently proposed by Woodward et al.12 The results should be independent of either of the models used if the separation is large enough. Using the nomenclature above, the interaction free energy for the pair and the resulting net force can be written as AA =
= -(aUAB/ab
ln (~xP[-UAB/~BTJ )&e
exp[-UAB/kBq
)&B/
(12)
(exp[-UAB/kBq
)&E
(13)
The average is now to be calculated over the noninteracting subsystems A and B, but the term in brackets contains an extra Boltzmann factor. This factor appears in both the numerator and denominator. The method relies on the fact that the interaction term is small, being treated essentially as a perturbation to the uncoupled systems. In practice, we simulated one subsystem and saved intermediate configurations in the Markov chain. Afterwards, the interaction free energy and the force are calculated by choosing pairs of configurations at random and placing them at the desired separation. One advantage in treating the weighting function, exp[-UAB/kBg, explicitly is that one can generate new configurations by rotating the colloids at fixed 6. These rotations do not have to be selected according to the MC process and lead to smaller fluctuations in our calculated forces.
Results and Discussion Below we show.how the intemlloidal force varies as a function of separation, salt concentration, polyelectrolyte flexibility, and colloid charge. In addition, we also present monomer distribution profiles and configurational properties for the polyelectrolytes. As we are presently only interested in qualitative effects, the parameters for the simulated systems have not been chosen to mimic any particular experimental situation. (a) Configurational Properties of the Polyelectrolyte Chains. Table I summarizes the calculated end-end and monomer-monomer separations for a few different parameter values. The average end-end and monomer-monomer (or bond) separations are defined as
(rmm2)I’*
= (cc(lr+l - r&2)”2/Ncb(Nm
- 1)
(14b)
where Nch is the number of chains (6 on each colloid) and Nm (equal to 1 I ) the number of monomers on each chain plus the point of attachment. The average configurational properties in (1 2) Woodward, C. E.; Jhsson, B.; Akesson, T.J. Chem Phys. lWIs, 89, 5145.
Interaction between Colloids and Carrying Polyelectrolytes
The Journal of Physical Chemistry, Vol. 95, No. 12, 1991 4823
TABLE I: Conflguntional Properties for Attached Polyelectrolyte chrinsa
5 5 5 5 5 5
IO 10
-60 -60 -40 -80 -40 -80 -60 -60
0.1 1.0 0.1 0.1 1.0 1.0 0.0 0.0
7.4 (8.8) 7.3 (7.5) 7.7 1.3 7.6 7.1 18.1 (23.6b) 21.6
24.0 (37)’ 23.2 (27) 25.4 23.8 25.5 22.2 31.4 (91b) 73.8
-100
-:
I
I
‘The values within parentheses are obtained for the corres nding free chain in isotropic salt solution. The colloidal radius is 18 Except for the last entry, where it is 60 A. bThese numbers were obtained at 0.001 M salt concentration.
Table I are obtained from simulations at large separations, so that we can essentially regard the system as two isolated colloids. A striking general observation is that both monomer-monomer and end-end separations are virtually independent of salt concentration. This can be contrasted with the behavior of free chains in an isotropic salt solution-the corresponding numbers are shown in parentheses. We also see small effects due to changing the colloidal charge. Note the total monomer charge was always held constant at +60. We can conclude that the configurational properties are largely determined by the external potential from the colloid and by the number of molecules per unit surface area. The polyelectrolyte monomers essentially move on the surface of the colloidal spheres, due to the strong adsorption potential. Here they sit at rather a high density, where the configurational properties are not expected to be markedly changed by variations in the long-ranged behavior of the interparticle interactions. In keeping with this assertion the end-end separation does not increase appreciably when we increase rdn from 5 to 10 A, though there is an increase in the average bond length. When we increase the colloid radius from 18 to 60 A, the average end-end separation increases due to the increase in available space on the surface and the reduced monomer-colloid interaction. At shorter separations between the colloidal particles we see small changes in the average monomer-monomer separations and a somewhat larger effect on the end-end separations. (b) Forces. (i) Comparison between Methods I and 11. Figure 6 shows a comparison of the two different methods (and models) for calculating the force. The distance, d, on the abscissa is the separation between the surfaces of the colloid particles. The discrepancy between methods I and I1 becomes most apparent at separations smaller than 10 A. This difference is due mainly to the different potential models used. Test calculations on model I show that the force curve is quite sensitive to the choice of the value of a. Model I1 does away with this ambiguity in the choice of parameters. The discrepancy is also partly due to differences in the relative accuracies of the two methods. Specifically, collisions of one colloidal particle with a monomer attached to the other particle are fairly rare, even at short separations; thus the TABLE 11: Variation of tbe Colloidal Force with Surface Charge Density and Polyelectrolyte Flexibility in a Salt-Free Solution‘ rmia= 5 A rmi, = I O A ri,, = IOA Rdl = 18 A RmI1= 18 A Rdl = 60 A d, A elec harm entr tot elec harm entr tot elec harm entr 1 -54 -88 136 -6 i 3 -49 -76 137 12 1.6 -36.5 5.5.5 2 -39 -84 93 -3Oi 3 -36 -70 95 -11 0.9 -44.4 52.3 -18 -66 56 -28 0.5 -36.2 47.4 4 -20 -80 54 -46 2 8 -6 -48 22 -32 i 2 -7 -54 30 -31 -0.3 -33.3 38.2 14 -1.2 -5.0 3.3 -2.9 h 0.5 -3 -40 16 -27 -0.4 -29.9 29.1 -0.3 -1.0 0.7 0.6 i 0.3 1 -20 9 -12 -0.4 -25.9 22.1 20 -0.2 -2.5 1.7 -1.0 -0.2 -16.3 12.1 34 -0.04 i O.Olb 44 -0.013 h 0.066 -0.1 -0.8 0.6 -0.3 -0.2 -10.4 7.6 -0.003 0.002b 0 -0.1 0.1 0 -0.1 -3.1 2.3 64
*
*
tot 20.6 8.8 11.7
5.2 -1.2 -4.2 -4.4 -3.0 -0.9
* d is the distance between the surfaces of the colloid particles. The errors for columns 2 and 3 range from 4.0 to 0.03 and 1.0 to 0.2 respectively. The force is given in units of lo-” N. bThese numbers were calculated by using the perturbation approach.
4824 The Journal of Physical Chemistry, Vol. 95, No. 12, 1991
Granfeldt et al.
TABLE III: Variation of tbe CoU0id.l Force with Addition of Simple EkctrolyteO c, = 0.0 M c, = 0.1 M
d, A
elec
I 2 4
-54 -39 -20 -6 -1.2 -0.4
8 14 19 "rmin= 5
harm -88 -84 -80 -48 -5.0 -0.7
entr
tot
136
-6
93 54 22 3.3 0.1
-30 -46 -32 -2.9 -0.4
elec -62 -40 -16 -2 1
1.2
harm
entr
-94 -88 -83 -45 -5 -1.4
148 91 52 19 2 0.6
tot
-8 + 2 -31 + 2 -41 f 4 -28 f 5 -2 f 1 0.4 + 0.3
elec -49 -25 -6 1 1
1
c,
= 1.0 M
harm
entr
-102 -95 -84 -54 -1 5 -6
165 105 61 29 9 4
tot 14+ 5 -I5 + 3 -29+ 3 -24 + 3 -si1
-112
A and RaII = 18 A. The force is given in units of IO-'* N.
to FeI.Figure 3 shows snapshots of Monte Carlo simulations for the gradual approach of the two colloids which clearly show the growth in the number of bridging chains with decreasing separation. Figure 6 and Table I1 also show how the force and its different components are affected b a change in the chain flexibility. Increasing rh from 5 to 10 jr results in a weaker bonding potential and therefore an increased monomer-monomer separation (see Table I). The main effect of this change is seen in Fhm. At the really short separations this term is slightly smaller than for rmi, = 5 A, but at intermediate separations the polyelectrolytes are still able to stretch across more easily than for the case rmi,= 5 A. Consequently, the harmonic term is much longer ranged and is significant even up to 60-Aseparation. Comparing the shape of the force curves for rmin= 5 and 10 A, we find the latter to have both a more shallow and a broader minimum. Consider now reducing the surface charge density on the colloids by increasing their radii. Table I1 shows how the force changes when the colloidal radii are increased from 18 to 60 A, while keeping rmi,= 10 A. The monomer-colloid attraction decreases and as a result the chains are more likely to bridge. On the other hand, the smaller attraction also leads to a decrease in the amount of stretching each bond experiences, with the result that Fhm is broader and more shallow. The decreased electrostatic attraction also leads to a marked decrease in Fel:indeed this contribution is essentially zero for the distances considered and the bridgin term is alone responsible for the attraction up to at least 60!I separation. As a comparison with the forces discussed so far we have also calculated the attractive van der Waals force acting between two hydrocarbon spheres of radius 18 A dispersed in a water-like medium; see the dotted curve in Figure 6. The van der Waals force was obtained from the expression9 W(r) = - A h ( x ) / l 2
h(x) = l / ( x 2 - 1 )
+ l / x 2 + 2 In ( 1 - 1 / x 2 ) , x = r/2RWll (15)
where A is the Hamaker constant, chosen as J. Except at very short separations we find that the attraction due to correlations and bridging is larger than the van der Waals interaction. (iii) Adding Salt: Varying Salt Concentration end Zd.In any real experimental situation the presence of salt is unavoidable and under many circumstances the salt plays an important role in the colloidal interaction. Under these conditions it is quite common that the charge of the adsorbed polyelectrolyte does not completely match the colloidal charge, Z,,,.The adsorbed amount of polyelectrolyte may either overcompensate or undercompensate2 , ; that is, the colloidal particle with its attached chains may not be electroneutral. Figure 7 shows how the addition of a simple electrolyte modulates the intercolloidal force. The overall shape and magnitude of the force curves are not greatly affected by salt. Increasing the ionic concentration tends to broaden the attractive region and lessen its magnitude. A comparison of the individual contributions (Table 111) shows that the bridging contribution tends to increase with the salt concentration though this difference is not large for separations less than 10 A. The salt reduces the potential barrier and the energy cost for monomers to %ave" the surface is smaller and hence we see a more long-ranged bridging contribution in the
0
10
20
Separation (A) Figure 7. Intercolloidal force as a function of separation for r& = 5 A, Rd = 18 A, and Zd = -60 at varying salt concentrations. The three curves are no salt (-), 0.1 M salt (---), and 1 M salt (---). presence of salt. The effect of added salt seems to be most prominent on the electrostatic term, which in general is reduced. Since electrostatic correlations will dominate at large separations, the reduction of correlations would here result in a weaker interaction for systems with salt. Figure 8a, b and Table IV show the force behavior when the colloidal particle carries a net charge. Surprisingly, even for these systems with effectively two similarly charged aggregates, the bridging is strong enough to cause a net attractive force. The electrostatic term is generally smaller than for an electroneutral system and is in fact in some cases repulsive at the minimum in the force curve; see Table IV. The lower surface charge when Zd = -40 leads to a long-ranged harmonic term while the opposite happens with Zdl = -80. For the latter case, the chains bridge only at very short separations and a slight increase in the separation causes the chains to literally collapse onto their respective surfaces. Under these circumstances we may expect the system to exhibit both a mimumum and a maximum in the force curve. The latter comes about as a consequence of electrostatic repulsion between nonneutral colloidal particles while the minimum is caused by bridging. Such a situation occurs for Z,,, = -80 and a salt concentration of 0.1 M (Figure 8a). In the 1 M case (Figure 8b) on the other hand, the strong screening of the electrostatic contribution causes the maximum to disappear. Figure 9 shows the monomer distribution functions for different colloid charges, ZWl,= -40, -60, and -80, respectively, at a separation of 14 A. The overcompensation of charge (Z, = -40) causes the polyelectrolyte chains to extend further out from the surface, while the opposite happens in an undercompensated system (Z,,, = -80). (c) Comparison with the Planar System. Finally, we compare the present results with those from simulations of two charged planar surfaces carrying grafted polyelectrolyte of opposite sign? We consider here only the case where there is no excess salt. In general, the same features of bridging and electrostatic correlations are present in both systems but differences appear due to the geometries. For both the planar and spherical cases the electrostatic contribution is generally attractive. As we noted before, the mean-field contribution to FeIis expected to be repulsive, which is not the case for the planar system where this contribution is zero. Thus
The Journal of Physical Chemistry, Vol. 95, No. 12, 1991 4825
Interaction between Colloids and Carrying Polyelectrolytes
TABLE I V Vahtioa of tk Colloidal Force with Different Degrea of Clurge Compeasrtioa in tbe Presence of Simple ElwtrdyteO c,
4A
Z ~ I
1 2 4 8 14 19
-40
-60
1
2 4
8 14 19 -80
1
2 4 8 14 19 @rh
elec
harm
c, = 1.0 M
= 0.1 M entr
tot
-34 -25 -10
-88 -80 -7 I
127 92
-1
-50
2 2 -62 -40 -16 -2
-1 3
21 8 3 148 97 52 19 2 0.6 165 97 44 12 1 0.1
1
1.2 -49 -18 14 24 18 11.0
58
-6
-94 -88 -8 3 -45 -5 -1.4 -105 -100 -8 3 -31 -2 -0.1
= 5 A and Rmll= 18 A. The force is given in units of
elec
harm
-1 1 -5 3 7 5 3 -49 -25
5f2 -13 & 2 -29 i 2 -24 A 2 -3 2 -1 f 2 -8 -3 1 -47 -28 -2 0.4 l l f 3 -21 3 -25 i 2 -1 f 3 17& 1 11 0.1
entr
-101 -93 -88 -63 -38 -1 9 -102 -95 -84 -54 -15
-6 1 1 1
-6
-72 -36 -8 0 0.3 1.5
tot 3 IO 3 -11 f 3 -12 f 4 -7 2 -2 2 14 -1 5 -29 -24 -5 -1 -5 4 -31 2 -44 4 -22 2 -1.2 & 0.5 0.9 f 0.6
142 108 74 44 26 15 165 105 61 29 9 4 174 101 48
-107 -96 -84 -40 -4.1 -1.3
30
*
**
18
2.6
0.7
N. I
'
I
I
I
0.2
ii
8
i
c 0
j
0.1
E
-40
c 0
I
z
' ,'
\
\
0
: I
-I 0.0 10
20
20
24
28
32
Position (A) Figure 9. Monomer distribution functions. The figure shows the number of monomers per unit length between the colloids. The separation is 14 A and parameters are as in Figure 8b.
Separation (A)
between two spheres gives a much smaller effect than with planes, due to their curvature and the ability of monomers to migrate out of the interstitial region. The overall effect is that the force in spherical systems seems to be attractive at much closer surface separations than with planes.
0
10
20
Separation (A) Figure 8. Intercolloidal force as a function of separation for r,, = 5 A, & = 18 A, and varying colloid charge. The cunm are Zd = -40 (-), Zdl = -60 (---) and Zdl = -80 (-*-); (a, top) in 0.1 M and (b, bottom) in 1.0 M salt solution. when compared with the planar case, we generally see a smaller electrostatic term, relative to other contributions to the force in the spherical system. In the planar case, the bridging force had a definite minimum at around the average monomer-monomer due to the separation. Closer approach sees a decrease in Fharm decrease in the amount of bond stretching and a saturation in the number of bridging bonds. This happens because all chains have the same distance to the second surface. With spherical particles, however, the bridging term decreases monotonically with distance. This is most likely associated with the increase in the reachable regions of low potential (near the colloid surfaces) as a given sphere approaches monomers attached to the other particle. The geometry also effects the repulsive part of the interaction. The repulsion is entropic in origin and reducing the separation
Conclusions Monte Carlo simulations have been used to study the interaction between two charged colloidal particles with adsorbed polyelectrolytes. The total force was discussed in terms of three contributions: a direct electrostatic interaction, a bridging attraction, and an entropically driven repulsion. For the case where the adsorbed charge due to chains equaled that of the colloids, the electrostaticterm was always attractive due to correlations between the ionic clouds around each of the colloidal particles. The bridging gave rise to a dominant attraction at short and intermediate separations and the entropic repulsion was only significant at very close ( I A) separations. On addition of a simple electrolyte the electrostatic term was reduced, due to screening of the interactions. The bridging was still significant and could even be enhanced at larger separations. In systems with a smaller colloidal charge than that of the adsorbed polyelectrolyte, the bridging proved strong enough to overcome the electrostatic repulsion and gave a strong attraction. In the case of a colloidal charge larger than the total polyelectrolyte charge, the bridging occurred at relatively shorter separations, and at larger separations chains were found to collapse on the respective surfaces. Although some variations are found in the force vs separation behavior for the investigated systems, the general features are similar. From the force curves an effective potential acting be-
J. Phys. Chem. 1991,95,4826-4832
4826
tween the colloids can be estimated. In doing so we find with decreasing separation, with only one exception, an increasing attraction down to a separation of 1 A. The depth of this attraction ranges from 4- to 8kBT. At shorter separations a repulsion ap-
pears. The exception is found for Zdl = -80 in a 0.1 M salt solution where the initial long-ranged potential is repulsive. Most likely these effective potentials, acting between colloids in a dispersion, would be strong enough to cause aggregation.
Elementary Surface Reactions in the Preparation of Vanadium Oxide Overiayers on Silica by Chemical Vapor Deposition Kei Inumaru, Toshio Okubara,* and Makoto Misono Department of Synthetic Chemistry, Faculty of Engineering, The University of Tokyo, Hongo, Bunkyo- ku, Tokyo 113, Japan (Received: September 28, 1990)
Elementary processes of the formation of vanadium oxide overlayers by the adsorption4ecomposition of VO(OC&), vapor on high-surface-area Si02were studied by IR (infrared) spectroscopy,TPD (temperature-programmed decomposition), and the stoichiometryof the surface reactions. In this study the adsorption4ecomposition process will be called a CVD (chemical vapor deposition) cycle. The structure of the vanadium oxide prepared by repeated CVD cycles was characterized by XRD (X-ray diffraction) and XPS (X-ray photoelectron spectroscopy). After the introduction of VO(OC2H5)3onto the Si02 at 423 K, the 1R peak due to the surface Si-OH groups disappeared, and the number of ethanol molecules evolved agreed with that of the surface Si-OH, indicating that all the Si-OH groups reacted with VO(OC2H&. On the basis of the stoichiometry of the gas-phase products and the vanadium atoms, it was confirmed that two surface species, Si-O-VO(OC2H5)2 (1) and (Si-0)2-VO(OC2HS) (2) were formed, the fraction of species 2 being 0 . 7 2 4 8 2 and 0.57-0.59 for Si02pretreated at 523 and 773 K, respectively. IR and TPD revealed that these species decomposed upon heat treatment to form vanadium oxide through the formation of V-OH and ethylene. The XPS peak ratios of V to Si as well as the XRD data showed that repeated CVD cycles gave highly dispersed vanadium oxides on SiOz as compared with that prepared by an impregnation method, especially at high loading levels.
Introduction
Recently, thin films of metal oxides dispersed on oxide supports have attracted much attention as catalysts. By the formation of oxide thin film, not only an increase in the surface area of the oxide overlayer but also the generation of active sites having novel functions is expected.' Vanadium oxide is interesting in this respect, as the catalytic activity of supported vanadium oxide has been reported to be sensitive to its microstructure. Mori et al. classified various oxidation reactions into structure-sensitive and -insensitive reactions on the basis of the activity per one V = O on the surface.2 Oyama et al. observed that the selectivity of ethane oxidation over V20s/Si02 depended strongly on the vanadium loadings, while that of ethanol oxidation was insen~itive.~ According to Wachs et al., V20s monolayers supported on Ti02 showed a high selectivity for oxidation of o-xylene.' Therefore, control of the microstructure may be necessary if the catalytic performance of supported vanadium oxide is to be improved. Chemical vapor deposition (CVD) is a useful method for the preparation of highly dispersed oxide Here, we call the process of deposition using reaction between surface sites such as OH groups and vapors of metal compounds a CVD methodeg Bond et al. found that a V205/Ti02catalyst prepared by the reaction between VOC13 vapor and surface OH groups showed a high selectivity in the oxidation of o-xylene to phthalic anhydride.5 We reported previously that a V2Os overlayer on S i 0 2 obtained from VO(OC2H5)3 vapor was highly dispersed.'O Besides these examples, there are several reports on CVD preparations of solid acid catalysts," and on AES" and analyses of oxide overlayers. In order to control the structure of V205overlayers formed by CVD, the elucidation of its elementary reactions during CVD is necessary. Kijenski et al. studied the reaction between VO(0C4H& and the surface OH group of oxides and observed that the numbers of V atoms deposited were nearly equal to those of the surface OH groups in the cases of A1203and Si02.13 How* T o whom correspondence should be addressed.
ever, elementary reactions during the adsorption and deposition are still obscure. In the present study, we have tried to elucidate the elementary surface reactions in the CVD process using VO(OC2H5)3 and the structure of vanadium oxide overlayers on SiOz by using XPS, XRD, IR, and TPD. Experimental Section Materials. Si02(Aerosil 200; 203 m2 g-I) was calcined at 773 K for 5 h in air and was stored at room temperature. The density of OH groups on the surface of the S i 0 2 was measured by two methods. The first was a titration method using sodium naphthalene as described in the 1iterat~re.l~The second was a H, D exchange between Si-OH and (CD3)&O, which is similar to a method described by Larson et aLI5 After S O z (about 0.3 g)
(1) Bond, G. C.; Flamerz, 1989, 87, 65.
S.;Shukri, R. Faraday Discuss. Chem. Soc.
(2) (a) Mori, K.; Miyamoto, A.; Murakami, Y. J. Phys. Chem. 1984,88, 2735. (b) Mori, K.; Miyamoto, A.; Murakami, Y. J . Phys. Chem. 1985,89, 4265. (3) Oyama, T. S.;Somorjai, G. A. J . Phys. Chem. 1990, 94, 5022. (4) Wachs, I. E.; Saleh, R. Y.; Chan, S.S.; Chersich, C. C. Appl. Caral. 1985, 15, 339. (5) (a) Bond, G. C.; Brcckman, K. Faraday Discuss. Chem. Soc. 1981, 72, 235. (b) Bond, G. C.; Kbnig, P. J . Caral. 1982, 77, 309. (6) Niwa, M.; Hibino, T.; Murata, H.; Katada, N.; Murakami, Y. J . Chem. Soc., Chem. Commun. 1989, 289. (7) Imizu, Y.; Tada, A. Chem. Lett. 1989, 1793. (8) Sato, S.;Urabe, K.; Izumi. Y. J . Catal. 1986, 102, 99. (9) Niwa, M.; Kato, M.; Hattori, T.; Murakami, Y. J . Phys. Chcm. 1986, 90, 6233. (IO) Inumaru, K.; Okuhara, T.; Misono, M. Chem. Lerr. 1990, 1207. (11) Ohhara. T.; White, J. M. Appl. Surf, Sci. 1987, 29, 223. (12) Jin, T.; Okuhara, T.; White, J. M. J . Chem. Soc., Chem. Commun. 1987, 1248. (13) Kijenski, J.; Baiker, A.; Glinski, M.; Dollenmeier, P.; Wokaun, A. J . Caral. 1986, 101, 1. (14) Kijenski, J.; Hombek, R.; Malinowski, S. J . Carol. 1977, 50, 186. (15) Larson, J. G.;Hall, W. K. J . Phys. Chem. 1965, 69, 3080.
0022-365419 1 12095-4826SO2.5010 0 1991 American Chemical Society
