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Langmuir 1998, 14, 3620-3624
Role of Hydration and Conformational Changes in Adsorption Dynamics of Ethyl(Hydroxyethyl)cellulose at the Air/Solution Interface B. V. Zhmud,* E. Poptoshev, and R. J. Pugh Institute for Surface Chemistry YKI, Box 5607, Stockholm SE-11486, Sweden Received October 10, 1997. In Final Form: March 16, 1998 Previous studies have discussed the kinetics involved in the reduction in surface tension of ethyl(hydroxyethyl)cellulose (EHEC) at the air/aqueous solution interface in terms of configuration changes involved in the transfer of individual polymer segments between soluted and adsorbed states. It was suggested that the process kinetics were governed by a substantially high activation barrier. In the present study, based on molecular mechanic simulations, it is shown that a nearly activation-free path between configurations exists for EHEC due to the extreme flexibility of the polymer chain. Hence, the relatively slow kinetics of the process should rather be attributed to a stepwise transformation between conformations which is expressed as a sequence of elementary processes involving a considerable number of intermediate isomers. This occurs over an extended time period. Also, the configuration changes involved in the uncoiling of the polymer at the interface are related to the hydration effects.
Introduction Several recent experimental studies have been reported on the dynamic surface tension behavior of ethyl(hydroxyethyl)cellulose (EHEC).1-3 In all cases, the time dependence of the surface tension of dilute EHEC solutions (determined over periods of several hours) shows three distinct consecutive kinetic regions which indicate the existence of the adsorption and reconfiguration stages in the adsorption process. Initially, an induction region occurs, followed by a fast-fall region where surface coverage occurs fairly rapidly and, finally, a mesophase region. Although the initial transportation of polymer from bulk solution to the subsurface is diffusion-controlled (dependent on molecular weight and bulk concentration), it was shown that the onset of surface tension change was explained by an increase in the number of adsorbed polymer segments in the interface following the uncoiling and spreading of the polymer chains. The conformation of the molecules must clearly change following the adsorption step if the molecules spread out on the surface. It has been suggested that the rate of this process will be dependent on the configuration changes involved in the transfer of individual polymer segments between the adsorbed state and the aqueous phase and that this process will be controlled by an activation barrier which could explain the kinetics of the process. In fact, this explanation, which indicates that the time dependence of surface tension is reflected by the number of adsorbed polymer segments, is in general agreement with the theoretical approach proposed by Scheutjens and Fleer.4 Another interpretation of the induction period in the relaxation of surface tension was given by Lin et al.,5 who pointed out the possibility of a gaseous-to-liquid-phase transition in the adsorbed layer: as long as such a transition is lasting, the addition of new molecules to the adsorbed phase will not be accompanied by a change in the surface pressure. * Corresponding author. (1) Poptoshev, E.; Um, S.-U.; Pugh R. J. Langmuir 1997, 13, 3905. (2) Um, S.-U.; Poptoshev, E.; Pugh, R. J. J. Colloid Interface Sci. 1997, 193, 41. (3) Nahringbauer, I. J. Colloid Interface Sci. 1995, 176, 318. (4) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (5) Lin, S.-Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055.
In the present study, we take an alternative approach to explain the process kinetics. From molecular mechanic simulations it is shown that, due to the high flexibility of the EHEC polymer chain, there always exists a relatively low activated path between configurations. In fact, a transformation of the type random coil-linear chain occurs in most cases through rotation of adjacent anhydroglucose links around bridging C-O-C bonds. Hence, for a large flexible EHEC molecule, a transformation between conformations can be represented as a sequence of elementary processes involving intermediate rotational isomers. Despite the low activation barriers separating these isomers, it can be anticipated that, with a sufficiently large number of intermediate stages, the configuration changes can occur over an extended period of time. Finally, from the molecular simulations performed with the EHEC molecule, an attempt is made to relate the configuration changes occurring during the uncoiling process to the summed effect of polymer bending and hydration. Among other things, this allows evaluation of the density of link packing and the gyration radius of EHEC for different states of expansion. Relation between EHEC Adsorption and Surface Tension Dynamics The adsorption of polymer at the gas/liquid interface and the redistribution of dissolved polymer within the adjacent superficial region of the solution is governed by the following set of equations:6,7
∂2c ∂c )D 2 ∂t ∂x D
∂c dΓ | ) ∂x x)0 dt
c(x,0) ) c0; Γ(0) ) 0
(1)
(6) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189. (7) Filippov, L. K.; Filippova, N. L. J. Colloid Interface Sci. 1997, 187, 352.
S0743-7463(97)01115-3 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/02/1998
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where c ) c(x,t) is the polymer concentration at position x at time t (x > 0, t > 0), c0 is the initial polymer concentration, D is the diffusion coefficient, and Γ ) Γ(t) is the adsorption at the liquid/gas interface (plane x ) 0). The constraints imposed on the solution express the polymer flux continuity requirement. The liquid phase is assumed to be semi-infinite, in which case it is implied that c(∞,t) ) c0. Integration by t leads to another equivalent formulation
Γ(t) )
∫0
t
D
∂c ∂x
|x)0 dt )
∂2c
∫∫∂x ∂t x,t
γ(t) ) γ0 - RT
dx dt )
∫0∞[c0 - c(x, t)] dx
(2)
that reflects the mass conservation.8 For the sake of simplicity, D is assumed to be independent of x despite the possibility that polymer concentration may differ with x. Two possible limiting types of adsorption behavior can be considered: If the adsorption equilibrium is achieved very fast, then the rate of the overall process is controlled by diffusion. In this case, covered in the pioneering work of Ward and Tordai9 and extensively studied thereafter,6,7,10,11 the adsorption equilibrium becomes settled in a time much shorter than that needed for any significant change in the concentration profile. Strictly speaking, this is the only case where the concept of dynamic surface tension allows a meaningful thermodynamic substantiation. The boundary condition in eq 1 will be different: c(0,t) ) f -1 0 Γ(t), where f(c) describes the adsorption isotherm, whichs because of extremely slow propagation of concentration changessdepends only on the local polymer concentration in the vicinity of the interface. Conversely, if the rate of diffusion is so fast that the concentration gradient caused by adsorption decays for a time much shorter than that needed to enable any noticeable change in Γ(t), the concentration of polymer appears to be constant throughout the solution phase, and the adsorption is said to be purely activationcontrolled. This can only happen in finite-size systems. For a semi-infinite system, unless there is no adsorption at all, c(0,t) * c(L,t) if L ) O(xDt), irrespective of the magnitude of D. Thus, depending on the observation time and the characteristic system dimensions, the same process can reveal either activation-controlled or diffusioncontrolled features. If a solution to eq 1 is found, the dynamics of surface tension can be deduced using the Gibbs equation12
dγ ) -Γ dµ ) -ΓRT
dc c
hence, ill-defined from a thermodynamic viewpoint, quantity.13 A common justification for using the dynamic surface tension data is that, on the time scale reflecting the characteristic solvent relaxation time, the evolution of the film can be a sufficiently slow process, which implies a mechanic rather than adsorption quasi-equilibrium to exist. Consequently, time can still be thought of as a parameter and handled appropriately. Integration of eq 3 yields
(3)
where µ is the chemical potential and R is the gas constant. It should be emphasized that all quantities in the latter equation refer to the equilibrium state of the system, whereas what is measured in the dynamic surface tension experiments may happen to be a nonequilibrium, and, (8) Frank-Kamenetskii, D. A. Diffusion and Heat Transfer in Chemical Kinetics; Nauka: Moscow, 1967. (9) Ward, A.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (10) Fainerman, V. B.; Makievski, A. V.; Miller, R. Colloids Surf. 1994, 87, 61. (11) Liggieri, L.; Ravera, F.; Passerone, A. Colloids Surf. 1996, 114, 351. (12) Davies, J. T.; Rideal, E. K. Interfacial Phenomena, 2nd ed.; Academic Press: New York, 1963.
∫c(t)c Γ dcc 0
(4)
where γ0 is the tension of the so-called zero-age surface. Since the composition of the latter is supposed to be identical with the composition of the bulk solution, the value of γ0 can be expected to be slightly different from that for pure water. For small t, assuming the first-order adsorption kinetics, one has
Γ(t) = kc0t (t , Γm/kc0)
(5)
where k is the adsorption rate constant and Γm is the limit adsorption corresponding to a completed molecular monolayer. On this time scale, the real-time limits of which depend on the magnitude of k, the adsorption process appears to be activation-controlled. To the moment t, changes in polymer concentration extend to the depth h ) (DΓm/kc0)1/2. Within this region
h dc ) -dΓ ) kc0 dt
(6)
If the system dimensions were much less than h, diffusion would effectively equalize the polymer concentration throughout the system and the overall process would appear to be activation-controlled independent of time. On substitution dc from eq 6 and Γ(t) from eq 5 into eq 4, the following asymptotic formula follows:
γ(t) ) γ0 -
c0RTk2 2 t (t f 0) 2h
(7)
It should be noted that, in the case of diffusion-controlled kinetics, a square root rather than a squared dependence on time should be expected in the above equations.10,14 Figure 1 compares the experimentally observed surface tension dynamics in the limit of low polymer concentration with that calculated from eq 7. Despite evident limitations, eq 7 leads to the following important qualitative conclusions that are in agreement with experiment: (1) The rate of changing γ(t) with time is proportional to the initial polymer concentration and temperature, and is an ascending function of the adsorption rate constant.1 (2) There is an induction period (since dγ/dt ) 0 as t ) 0) on the dynamic surface tension vs time curve, followed by a fast-fall region. This cannot be explained in the framework of diffusion-controlled kinetics. Similar behavior is also observed in surfactant solutions.5,15 For large t, the case not covered in the present study, the dynamic surface tension should approach its static limit corresponding to the equilibrium adsorption. This (13) Defay, R.; Petre, G. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1971; Vol. 3, pp 27-81. (14) Eastoe, J.; Dalton, J. S.; Rogueda, P. G. A.; et al. J. Colloid Interface Sci. 1997, 188, 423. (15) Ravera, F.; Ferrari, M.; Liggieri, L.; et al. Langmuir 1997, 13, 4817.
[
3622 Langmuir, Vol. 14, No. 13, 1998
dcn ) kn-1cn-1 dt dcn-1 ) kn-2cn-2 - kn-1cn-1 dt ............................. dc1 ) -k1c1 dt
]
Zhmud et al.
(9)
The latter is easily handled with the Laplace transform method, leading to the following expression for cn(t):
c (0) k k ...k
∫0∞cn(t) exp(-st) dt ) s(s + k 1)(s +1k2)...(sn-1+ k 1
2
n-1)
(10) Figure 1. Experimental time dependence of the surface tension of an aqueous solution containing 2.5 ppm EHEC.1 The solid line is the best-fit parabola (γ0 ) 7.22 × 10-2 N m-1; c0RTk2/2h ) 1.86 × 10-9 N m-1 s-2).
accounts for departure of the last experimental points from the parabola calculated according to eq 7. Configuration Changes in the Adsorbed Layer Although eq 2 alone would explain the existence of an initial plateau in time dependence of the dynamic surface tension from phenomenological positions, the complexity of the internal structure of polymer molecules is herewith neglected. However, when taken into account, the structural changes in polymer molecules can suggest another explanation for this fact. In a recent paper by Poptoshev et al.,1 the observed increase in the foam film stability in the course of time has been attributed to configuration changes of the adsorbed EHEC polymer. The authors suppose that the low rate of this process is explained by its high activation energy. On the contrary, the molecular mechanic simulations undertaken in the present study suggest that because of extreme flexibility of the polymer chain there should always exist a nearly activation-free path between different configurations. In particular, a transformation of the type random coil-linear chain in most cases can be effected through rotation of adjacent anhydroglucose links around bridging C-O-C bonds. The activation energy of such a process is relatively low. The role of chain flexibility as a factor governing the preferable configuration of adsorbed macromolecules was also emphasized by Fainerman et al.16 There is another explanation of why the conformational changes are so slow. For a huge polymer molecule, a transformation between any two configurations can be represented as a sequence of elementary processes involving a lot of intermediate rotational isomers k1
k2
kn-1
c1 98 c2 98 ...cn-1 98 cn
(8)
where ci is the concentration of the ith isomer, ki is the corresponding rate constant, and n is a sufficiently large integer number. According to the classical first-order kinetics, ci and ki are inter-related by the following system of equations: (16) Fainerman, V. B.; Miller, R.; Wustneck, R. J. Colloid Interface Sci. 1996, 183, 26.
The behavior of cn(t) for t f 0 can be concluded by making s f ∞ and inverting the resultant limiting form of eq 10.
tn-1 (t f 0) cn(t) = c1(0) k1k2...kn-1 (n - 1)!
(11)
In this limit, (i) the activation energies of all the intermediate processes are summed, and (ii) a certain induction period occurs before cn(t) starts to increase. It is of interest to estimate the duration of the caused delay. This can be done as follows. Assume, for the sake of simplicity, k1 ) k2 ) ... ) kn-1 ) K. Then
cn(t) )
c1(0) Kn (n - 2)!
∫0ttn-2 exp(-Kt) dt
(12)
In the latter expression, the integrand has the maximum at t ) (n - 2)/K. This suggests conclusively that for a sufficiently large number of intermediate stages, as in the case of flexible polymer chains, the initial delay can be infinitely long independent of how high the activation barrier for the sequential steps in eq 8 is. This also applies to long-chain surfactants.5,15 Now, if it happens by whatever reason that only a particular chain configuration is favorable for adsorptionsand changes in configuration caused by adsorption are the reality, indeedsthis delay with generation of a sufficient amount of molecules having the required configuration in the vicinity of the interface will inevitably result in an induction period in the surface tension dynamics. Hydration Effects For noncharged macromolecules, it is the hydration energy that is responsible for the preferable configuration the polymer adopts in the solution.17 It is well-known that hydrophilic macromolecules tend to accept an expanded conformation, whereas the hydrophobic ones tend to contract or to form aggregates composed of several macromolecular units. Hydration also affects the spreading of surface-active compounds on surfaces.18 The next section is intended to give a simple semiquantitative explanation of such a behavior on the basis of the results of molecular mechanic simulations performed with the EHEC polymer. These simulations were performed using real-size polymer chains composed of 40-400 anhydroglucose units and having molecular weights up to approximately 100 000, the latter being the average mo(17) Munk, P. Introduction to Macromolecular Science; John Wiley: New York, 1989. (18) Steinbach, H.; Sucker, C. Adv. Colloid Interface Sci. 1980, 14, 43.
Adsorption Dynamics of Ethyl(hydroxyethyl)cellulose
Langmuir, Vol. 14, No. 13, 1998 3623
Figure 3. Three main contributions to the hydration energy of different configurations of an EHEC macromolecule with Mw ) 103 802 as functions of the radius of gyration. Table 1. Characteristics of the Most Probable Hydrogen Bonds between Water and Polar Groups of EHEC Polymer (Calculated by MNDO-PM3) bond type CH3CH2
Figure 2. Computer-generated random-coil conformation of an EHEC macromolecule having a molecular weight Mw ) 103 802 and a radius of gyration of 6.8 nm.
O
(CH2)2OCH3
d(OH), nm
∠(HO...H), grad
-∆Hf, kcal mol-1
0.183
172.1
3.16
0.182 0.181
177.5 166.6
2.67 3.28
0.183
167.5
3.76
0.180
173.6
1.57
HOH CH3O(CH2)2OH CH3O(CH2)2
OH2
O H HOH
lecular weight of polymer used in the work of Poptoshev et al.1 The polymer chains have been built by translation of a tetramer elementary link in a random manner, implementing the so-called “random flight” idea.17,19 The molecular structure of the link was essentially the same as that accepted in ref 1, except the geometry pre-optimized with the MM2 method.20 Figure 2 represents one of the generated conformations. The hydration energy of a polymer can be evaluated by composing the energy balance of three main components, viz., the energy of polymer-water interaction, the energy of intramolecular interactions within the polymer molecule, and the energy of solvent cavitation. The first component was calculated by scaling the energy of interactions between a single anhydroglucose link and a few water molecules to the desired polymer size, using as the scaling factor the ratio of corresponding solventaccessible surface areas. The degree of hydration was taken in accordance with the results of Carlsson et al.21 For a linear macromolecule with Mw ) 103 802 and chain length 168 nm, this yields about -15 000 kcal mol-1, or -150 kcal mol-1 per link. The latter result was obtained by summing the energies of the most feasible hydrogen bond between water and polar groups of the link. The bond energies were calculated in the framework of MNDO-PM3;22 a brief summary is given in Table 1. Besides listed therein, there are weak hydrogen bonds between aliphatic hydrogens and water. The energy of the latter does not normally exceed 1 kcal mol-1. The second component in the balance reflects the existence of intramolecular strains and link-to-link interactions. It is very susceptible to conformation changes. (19) Nakata, M. Polymer 1996, 38, 9. (20) Lipkowitz, K. B. QCPE Bull. 1992, 12, 6. (21) Carlsson, A.; Lindman, B.; Nilsson, P.-G.; Karlsson, G. Polymer 1986, 27, 431. (22) Stewart, J. J. P. J. Comput. Chem. 1989, 10, 209.
OH O
HOH OH
OH OH OH O OH OH
OH
OH2
To evaluate it adequately, a molecular mechanic simulation of a 20-links-long EHEC chain in different conformations was performed using the atom-atom potential approximation23 with the extended MM2 force field,20 followed by scaling the results to the desired polymer size. The last component in the energy balance is the cavitation energy. It accounts for the work required to remove a sufficient number of solvent molecules to free the room for accommodation of a huge solute molecule. This amount of work can roughly be estimated as the product of the solvent surface tension by the solvent-exposed surface area of the solute. For a linear EHEC macromolecule with Mw ) 103 802, the solvent-exposed surface area is estimated to be 894.9 nm2, thus leading to a cavitation energy of 9330 kcal mol-1. At this point, it should be noted that the terms “solvent-exposed surface area” and “solvent-accessible surface area” have different meanings,24 and the former is much less susceptible to conformation changes than the latter. For this reason, the ratio of cavitation energies for two different conformations was taken to be equal to the ratio of their solvent-accessible surface areas. The results of the above-mentioned calculations are summarized in Figure 3. One may notice a small minimum in the total energy curve, which represents the (23) Pertsin, A. J.; Kitaigorodsky, A. I. The Atom-Atom Potential Method; Springer-Verlag: Berlin, Germany, 1987. (24) Hawkins, G. D.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. 1997, 101, 7147.
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Zhmud et al.
energy, or better said, the heat of hydration. This indicates a higher probability of coiled conformations to exist. Entropic effects can produce even a deeper minimum on the corresponding free energy curve. Unfortunately, there is no clear way to evaluate these effects from first principles. This is where parametrized models can come in handy.24 For the sake of comparison, we have performed calculations of the free energy of hydration of a linear conformation of EHEC by two different methods, viz., the method of geometry-dependent atomic surface tensions with implicit electrostatics developed and implemented in the OMNISOL program by Hawkins et al.,24 and later rationalized and ported on a PC platform by one of us, and molecular dynamic simulation25 of a 3-links-long chain hydrated by 1000 water molecules. The results are -58.5 kcal mol-1 per link according to the first method and -64.0 kcal mol-1 per link according to the second method. Hence, the hydration free energy of the whole macromolecule in the linear-chain conformation should fall in the range -5850 to -6400 kcal mol-1, which is not much different from the corresponding heat of hydration. Apparently, despite a good correlation in the results obtained, the fact remains that neither of the above models adequately evaluates the configuration entropy.26 A closer inspection of the results shown in Figure 3 allows a simple explanation of the configuration changes caused by hydrophobic substitution or adsorption at the air/water interface. For example, the hydrophobization
increases the cavitation energy without a major effect on the two other components in the energy balance. Consequently, the minimum on the total energy curve becomes deeper and is shifted to more compact configurations. In the adsorption process, as a polymer molecule is expelled from the bulk to the surface, it tries to keep its polar groups oriented to the water phase, a kind of behavior typical of surfactants, whereas nonpolar groups are oriented to the air, thus entailing a reduction in the cavitation energy. This could be represented by scaling down the cavitation energy curve in Figure 3. As a result, the minimum on the total energy curve is shifted to larger gyration radii and the potential barrier preventing the molecule from expansion is reduced and the slope becomes less severe. This suggests an increasing probability for the polymer molecule to extend itself over the surface, as in accord with the conclusions drawn before from film stability measurements.1 The latter effect should be expected to manifest itself even stronger in the case of hydrophobically substituted EHEC, thus making it easy to understand the origin of its enhanced surface activity.2,3,27
(25) Clementi, E.; Corongiu, G.; et al. In Modern Techniques in Computational Chemistry: MOTECC-90; Clementi, E., Ed.; ESCOM: Leiden, 1990; pp 805-888.
(26) Karplus, M.; Kushick, J. N. Macromolecules 1981, 14, 325. (27) Thuresson, K.; Karlstrom, G.; Lindman, B. J. Phys. Chem. 1995, 99, 3823.
Acknowledgment. The work of B.V.Z. and E.P. was partly supported by the Swedish Institute and the Royal Swedish Academy of Sciences. LA971115S
