摘要:

J. Chem. SOC.,Faraday Trans. 2, 1983,79, 1-7 Reaction of the Triplet State of Retinol with Oxygen BY GERALDJ. SMITH Physics and Engineering Laboratory, Department of Scientific and Industrial Research, Private Bag, Lower Hutt, New Zealand Received 26th October, 1981 The yield of singlet oxygen from the reaction of oxygen with the triplet state of retinol is 0.25 iz0.05. The low yield can be explained by the existence of three competing pathways for the reaction of the triplet state of retinol with oxygen: (a)energy transfer: 'T* + 3~2'T+ lo2* (6) charge transfer: 3T*+302-+ 'T+302 (c) spin exchange with a triplet state in a twisted conformation: 3P*+ 302-+pT+(1-p)C+ 302. The rates of isomerization of all-trans retinol sensitized by triphenylene were measured in the presence and absence of oxygen.From the ratio of these rates and the yield of singlet oxygen it is concluded that the disappearance of triplet state via pathway (c) is much slower than the combined rates via pathways (a) and (6). Also, the rate of path (6) is 2.7 times faster than path (a). Theoretical considerations show that path (6) is probably mediated by charge-transfer interactions between the all-trans triplet state of retinol and oxygen. Triplet states can react with oxygen by two pathways; uiz.: (i) energy transfer: 3M*+ 302-b1M+ '02* (ii) enhanced intersystem crossing: 3M* + 02+'M + 302. The rate constant for the reaction of oxygen with the triplet state of retinol in hexane' is 4.7x lo9dm3 mol-' s-'. The rates of reaction of triplet-state species with oxygen in which energy transfer is involved are limited2 by diffusion and a spin-statistical factor of 1/9.The maximum rate constant measured for these reactions in hexane3 is 3.3 x ill9dm3 mol-' s-', and since the rate constant for the triplet state of retinol is greater than this it is concluded that intersystem crossing, which has a spin-statistical factor of 1/3, is participating in the reaction. Although intersystem crossing has a greater spin-statistical factor than energy transfer, the latter pathway generally predominates because of relative magnitudes of the Franck-Condon factors. However, on occasions, certain processes such as charge transfer (3M* + 02+[M"---O;-]) or geometric changes of the triplet state (isomerization) can promote intersystem crossing.Satiel et al.4 have found that enhanced intersystem crossing is involved in the reaction between the triplet state of stilbene and oxygen. Stilbene, like retinol, has a double bond which can twist and as the angle of this twist approaches 90°, the energy of the ground state rises to a maximum, while the energy of the triplet state falls slightly. In this configuration the potential energies of the two states are close together or cross and as a consequence the Franck-Condon factor is high. Intersystem crossing, promoted by interaction with the triplet, ground state of oxygen, is therefore rapid. Since enhanced intersystem crossing is subject to a spin-statistical factor of one third, the rate of this process can be as much as three times faster than the oxygen quenching process involving only energy transfer.Retinol has a conjugated system of double bonds, and twisting is possible about some of these double bonds which changes the potential energies of the electronic 1 REACTION OF RETINOL WITH OXYGEN states.5P7 The mechanism described for stilbene could therefore explain the com- paratively high rate constant for the reaction of oxygen with the triplet state of retinol. Enhanced intersystem crossing, mediated by a charge-transfer interaction with oxygen, can also occur when the triplet-state molecule has a low oxidation potential. This type of reaction has been suggested by Garner and Wilkinson8 for the quenching of the triplet states of some amines and ketones by oxygen.Of the pathways for oxygen quenching of triplet states considered above, only energy transfer produces singlet oxygen, whilst intersystem crossing promoted by geometric changes leads to trans to cis isomerization. It is therefore possible to estimate the contributions of these pathways to the quenching of the triplet state of retinol by oxygen by studies of singlet-oxygen yield and photoisomerization. The following experiments were conducted with this objective. EXPERIMENTAL MATERIALS B.D.H. spectroscopic-grade hexane was used as the solvent. Grade I all-trans retinol from the Sigma Chemical Co and triphenylene supplied by Koch-Light were used. All solutions were prepared immediately before use under a Kodak 1A red safety light and the experiments were also performed under this lighting to avoid unwanted photolysis of retinol.The solutions were bubbled with oxygen + nitrogen gas mixtures of various compositions to produce concentrations of oxygen ~1.4 mol dmp3. The concentrations of oxygenx present in the solutions were determined by measuring the rates of decay of the triplet state of anthracene and taking the rate constant for the reaction of the triplet state of anthracene with oxygen in hexane as 3.33 x 10’ dm3 molpl s-1.3 For measurement of the quantum yield of singlet oxygen, the triplet state of retinol was produced by photolysis of a 1x mol dm-3 solution of retinol containing 8 x loP5 rnol dmP3 triphenylene as a sensitizer and 1.4 x 10 mol dm-3 oxygen with a 20 ns pulse of 264 nm light from a quadrupled neodymium laser.The laser light was attentuated to avoid ground-state depletion. The flash-photolysis apparatus has been described fully elsewhere.’ The solution also contained 1x lop5mol dmp3 diphenylisobenzofuran, DPBF, which reacts rapidly with singlet oxygen, and the extent of bleaching of DPBF at 415 nm was measured. The quantum yield of singlet oxygen from the reaction of the triplet retinol with oxygen was then determined by comparing this bleaching with that obtained by photolysing an air- saturated anthracene solution containing DPBF with an absorbance of 1.6 at 264 nm in a 1cm cell. At the concentrations employed in this work, triphenylene absorbs 76% of the light absorbed by the mixture.In the flash-photolysis apparatus the analysing beam intersects only the first 1mm of photolysed solution. Thus the laser photolysing light absorbed by triphenylene in this part of the solution is approximately the same as that absorbed in the same part of the anthracene solution with an absorbance of 1.6 in a 1cm cell. Every reactive encounter of an oxygen molecule with an anthracene molecule in the triplet state gives a molecule of singlet oxygen.” However, the yield of the triplet-state anthracene in air- saturated hexane is calculated to be 0.81,’so the magnitude of the bleaching of DPBF by singlet oxygen produced by the reaction of oxygen with the triplet state of retinol was normalized accordingly to obtain the quantum yield of singlet oxygen from this reaction.ISOMERIZATION STUDIES For measurement of the rate of isomerization of all-trans retinol, the triplet state of retinol was again produced by reaction with the triplet state of triphenylene. Solutions G. J. SMITH TABLE RA RATE CONSTANTS FOR SOME OF THE REACTIONS OCCURRING IN A SOLUTION CONTAINING RETINOL, TRIPHENYLENE, DPBF AND OXYGEN WHEN IT IS IRRADIATED AT 264nm reaction rate constant ref. ~ 'Tri+hv -+ 'Tri" (1) 'Ret+hv -+ 'Ret" (2) 'Tri" + 'Tri 'Tri" + 0, + 'Tri" + 'Oo, (31 (4) 'Tri" + 'Tri" 'Tri" + Ret -+ 'Ret" + 'Tri (5) (6) k5=3x10*s k6s2 x 10"' dm' mol ' s ' 12 'Tri" + '02+ 'Tri + '0,or 'Oj (7) k, = 3 x 10' dm' mol-' spl 3 'Tri + Ret + 'Tri + 'Ret* 'Ret" + 302-+ 'Ret + '02or '02 (8) (9) k, = 1.5x 10'" dm3mol-' s-' k, = 4.7 x lo9dm3 mol ' spl 13 1 'Ret" -+ 'Ret 'Ret" + '0, -+ 'Ret* + '0, 'Ret" -+ 'Ret (10) (11) (12) k,,,= 7.9 x 10, S-l kll= 2 x 10"dm' mol k1,=2u10Xs ' ' spl 1 11 14 DPBF+ '0, -+ product (bleaching at 415 nm) (13) containing 1.5x lop4mol dm-' trans-retinol and 2 x mol dm-' of the photosensitizer in hexane were irradiated with light from a 200 W super-high-pressure mercury arc which was passed through filters of 5 cm pathlength NiSO, solution and 10 cm pathlength chlorine gas.The triphenylene absorbs >95% of the light incident on the solution with this source and filter combination. 50mm' samples were withdrawn from the solution at various times during the irradiation.The isomers were separated by high-pressure liquid chromatography (h.p.1.c.) using a Waters Associates Bondpak NH2 column. The eluent used was 50% chloroform+hexane and the flow rate was 2.0 cm' min-'. The relative amounts of the all-trans and all-cis isomers of retinol present in these samples were determined by measuring the areas under the h.p.1.c. peaks associated with the isomers. RESULTS AND DISCUSSION DETERMINATION OF THE QUANTUM YIELD OF SINGLET OXYGEN The quantum yield of singlet oxygen in the system studied was found to be 0.25 f0.05. This value was the mean of ten independent determinations and the uncertainty was the extreme deviation of the measurements from the mean. The following considerations show that the singlet oxygen is only produced in the reaction of oxygen with the triplet state of retinol.The intersystem crossing for retinol is very low" so triphenylene had to be used to photosensitize the formation of the triplet state of retinol. When a solution containing triphenylene (Tri), retinol (Ret) and oxygen is irradiated, a number of competing reactions are possible and these are presented in table 1 along with some of their rate constants and the reference which gives the rate constant or information from which it can be derived. REACTION OF RETINOL WITH OXYGEN 1.4 1.2 - L 1.0 t-l '5 0.8 0.6 0.4 0.2 I I I I 1 I L I 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 [02]-'/104dm3 mol-' FIG.1.-Rate of isomerization of all-trans retinol at various concentrations of oxygen compared with the rate of isomerization in the absence of oxygen. At the reagent concentrations present in this system, the possible reactions set out in table 1are dominated by the sequence, hu --+ 3Tri* --+ 3Ret*+02 -+Ret+'02* or 302. The low quantum yield of singlet oxygen from the reaction of the triplet state of retinol with oxygen means that energy transfer only makes a relatively minor contribution to the overall reaction. ISOMERIZATiON The ratios of the rates of photoisomerization of all-trans retinol at various oxygen concentrations to the rate in the absence of oxygen are shown in fig. 1. A possible reason for the low yield of singlet oxygen is that oxygen reacts very rapidly with the triplet state of the retinol molecule which is twisted about the 11-12 double bond.The electronic excitation energy associated with the triplet state is transferred into potential energy associated with torsion of the double bond and spin, in the triplet to ground-state transition, is conserved by interaction with oxygen. The potential energies for the ground and excited states of retinoids as a function of rotation about the 11-12 double bond have been calculated by Becker et d7 The relevant feature of this is that the energy of the lowest triplet-state level is nearly independent of rotation about the 11-12 bond and at angles near 90" it is close to the maximum in the ground-state level.The processes involved in the early stages of the photoisomerization of all-trans retinol can therefore be repre- sented by the following scheme: G. J. SMITH 5 3(~---02)* 3(~---02)* or A T or 3T* 3P*I'P \ 'C where T is the all-trans isomer, P is a twisted conformation and C is the 11-cis isomer of retinol. During the early stages of the irradiation, insufficient 11-cis isomer is present to significantly compete with the all-trans isomer in the sensitizing reaction with the triplet state of triphenylene, 3Tri. At this stage of the irradiation, the following reactions occur in the absence of oxygen: Tri + hv -+ 3Tri* 3Tri*+T + 'Tri + 3T* 3*-T .-3P* where a is the fraction of 'P which converts to 'T. When oxygen is present, the following reactions occur: 3T*+302-+ 'T+'02* 3T*+302-+ 'T+"02 'p*t302 + p'~+(i-p)'c+~o~ 3(~---02)** 3(~---02)* 3(T---02)* and 3(P---02)*are the encounter complexes formed between oxygen and 3T*and 3P*,respectively.Assuming steady-state kinetics for 3P*,3(P---02)*,3T*and 3(T---02)* provided k14[T]>> k7[02] (which is satisfied where [O,]6 1.4x mol dmP3). R is the ratio of the initial rates of disappearance of T in the absence and in the presence of oxygen, kA= (k,+kb), y = (1-p)/(l -a) and K = kd/k-d. REACTION OF RETINOL WITH OXYGEN Truscott et al.' have measured the rate constant for the disappearance of the triplet state of retinol with a T-T absorption at 400 nm in the absence and in the presence of oxygen.From the potential-energy diagrams for the electronic states of retinoids7 it is concluded that the retinol triplet state with a T-T absorption at ca. 400 nm must be the twisted form, 3P*.Therefore k16 = 7.9x lo4s-l and kA/K+ k, = 4.7 X lo9dm3 mol-l s-l. From these rate constants, the gradient and intercept of the graph shown in fig. 1and the yield of singlet oxygen, which is Y= ka = 0.25kAfKk, it is deduced that y = 1,i.e. the fraction of the twisted form of retinol which relaxes to the all-trans isomer is approximately the same with or without oxygen. It then follows that i.e. the rate of disappearance of triplet state via path (c) is slow compared with the combined rates of paths (a)and (b). Also ka=0.27 (ka+ kb).This means that only 27% of the reactive encounters of oxygen with the all-trans triplet states of retinol, 'T*, lead to the formation of singlet oxygen [path (a)]. The remaining all-trans triplet states of retinol which react with oxygen do so by an intersystem crossing process [path (b)] which could be promoted by charge-transfer interactions. Since ka s3.3 x lo9dm3 mol-' s-l (the maximum rate constant for an energy-transfer reaction between a triplet state and oxygen) it is concluded that K d2.75. These observations are in agreement with theoretical considerations which predict that charge transfer will play an important part in the quenching of 3T* by oxygen. The energy of a charge-transfer complex between retinol and oxygen is given by'' ECT =ERet/Ret+ -EO,/O, -c where ERet/~et+is the oxidation potential of retinol(O.95 V in THFI6), Eo;/o, is the reduction potential of oxygen (-0.82 V in CH3CN and -0.77 V in DMSO") and C is the stabilization energy of the ion pair, which in non-polar solvents at room temperature is ca.0.09 eV.'' Therefore ECT is ca. 1.66 eV compared with the triplet-state energies of the all-trans and the 90" twisted retinol, which are believed to be ca. 1.9 eV.7 CONCLUSIONS The reaction of oxygen with the triplet state of retinol goes by three pathways, viz. (a) energy transfer: 3T* + 302-+ 'T + lo2*. (6) charge transfer: 3T* + 302+3[T"---0i-]* --+IT + 302 G. J. SMITH 7 (c) spin exchange with a triplet state in a twisted conformation: 3P*+3023PT+(1-p)C+ 302.The charge-transfer pathway (b)is the most important. T. G. Truscott, E. J. Land and A. Sykes, Photochem. Photobiol., 1973, 17,43. L. K. Patterson, G. Porter and M. R. Topp, Chem. Phys. Lett., 1970, 7, 612. 0.L. J. Gijzeman, F. Kaufman and G. Porter, J. Chem. Soc., Faraday Trans. 2, 1973, 69, 708. J. Satiel and B. Thomas, Chem. Phys. Lett., 1976, 37,147.' J. R. Wiesenfeld and E. W. Abrahamson, Photochem. Photobiol., 1968,8, 487.'S. J. Formosinho, Proc. Lisbon Conf. on Excited States of Biological Molecules, ed. J. B. Birks (Wiley, London, 1976).' R. S. Becker, K. Inuzuka, J. King and D. E. Balke, J. Am. Chem. Soc., 1971,93, 43. ' A. Garner and F. Wilkinson, Chem. Phvs. Lett., 1977, 45, 432.'G. J. Smith, J. Chrm. Soc., Faraday Trans.2, 1982, 78, 769. 10 A. Garner and F. Wilkinson, Singlet Oxygen Reactions with Organic Cmipaunds and Polymers, ed. B. Ranby and J. F. Rabek (Wiley, New York, 1978).II T. A. Rosenfeld, A. Alchalal and M. Ottolenghi, Chem. Phys. Lett., 1973, 20, 291. 12 S. L. Murov, Handbook of Photochemistry (Marcel Dekker, New York, 1973).13 A. Sykes and T. G. Truscott, J. Chem. Soc., Faraday Trans. 2, 1971,67, 679. 14 T. Rosenfeld, A. Alchalal and M. Ottolenghi, Proc. Lisbon Conf. on Excifed States of Biological Molecules, ed. J. B. Birks (Wiley, London, 1976).15 H. Knibbe, D. Rehm and A. Weller, 2.Phys. Chem., N.F., 1967,56, 95. 16 Su-M. Park, J. Electrochem. Soc., 1978, 125,216. 17 M. E. Peover and B. S. White, Chem. Commun., 1965, 183. 18 H. Beens and A. Weller, MolecularLuminescence, ed. E. Lim (W. A.Benjamin, New York, 1969). (PAPER 1/1667)

ISSN:0300-9238    

DOI:10.1039/F29837900001

出版商:RSC    

年代:1983

数据来源: RSC

ISSN:0300-9238    

DOI:10.1039/F298379BX003

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1983,79, 9-17 Dielectric Dispersion Studies of some Unsaturated Compounds in Dilute Solutions GOWRI KRISHNA AND JANDHYALABY JANDHYALA SOBHANADRI* Department of Physics, Indian Institute of Technology, Madras 600 036, India Received 30th November, 1981 The dielectric permittivity and dielectric loss of solutions of four unsaturated triple-bonded com- pounds were measured as a function of frequency (10 kHz, 8.43, 11.3 and 35.6GHz), temperature (33-70 "C) and concentration (0.07-0.01 weight fraction). The data are analysed in terms of a double relaxation process. The most probable relaxation time, the overall molecular relaxation time and the group relaxation time were calculated. The dipole moments were also calculated along with the thermodynamic parameters.The results are discussed in terms of association effects in these compounds and the dielectric data are correlated with the infrared spectral data. The dielectric dispersion of polar liquids has been studied extensively over the last few years because of the availability of better experimental facilities over a wide frequency range, including the microwave frequencies. Since the recognition of hydrogen bonding, one of the systems studied in great detail has been the alcohols, where self -association has a marked influence on the physical properties.' The dielectric absorption of pure liquid aliphatic alcohols is characterised by three discrete relaxation processes.2 The problem is most pronounced for pure alcohols owing to the many possible modes of self-association and so efforts have been directed towards the study of solutions in non-polar solvents, where increasing dilution will inevitably reduce self-association.Dielectric permittivity and loss measurements for alcohols in non-polar solvents provide valuable information about the inter- and intra-molecular relaxation pro- cesses. Such studies are mostly confined to saturated alcohols, particularly the primary alcohol^,^ and not much has been done on the unsaturated alcohols either in pure liquid or dilute solution. In an earlier publication from this laboratory, the dielectric properties of double-bonded unsaturated compounds were rep~rted,~ and it was shown that there are two relaxation processes, as in the case of saturated primary alcohols, but of much greater magnitude. In the present paper the dielectric permittivity and loss measurements of four unsaturated triple-bonded compounds in solution are presented as a function of frequency, concentration and temperature.Three of these compounds are alcohols, where the effect of the OH group is seen. For the sake of comparison, a compound with Br in the place of the OH group is also studied. The dielectric data are coupled with infrared studies and an attempt is made to correlate the two. The purpose of this investigation is to study the effect of the OH group, which is present in the associated and unassociated species, on the dielectric relaxation. The dipole moment and its possible variation with concentration and temperature are also investigated.9 10 DIELECTRIC MEASUREMENTS ON UNSATURATED HYDROCARBONS EXPERIMENTAL The compounds selected for the study were: HC_C-CH,OH, propargyl alcohol, P I; HCEC-CHOH-CH,, 3-butyn-2-01, P 11; HC_C-CH,Br, propargyl bromide, P 111; and H,C-C'_C-CH,OH, 2-butyn-1-01, P IV. All these compounds were obtained from com- mercial sources and were dried over anhydrous sodium sulphate. Before conducting the experiments the compounds were subjected to fractional distillation and the following boiling points were obtained (literature' values in parentheses): P I, 113 "(3113.6 "C); P 11, 107.5 "C(107 "C); P 111,88.5 "C(89 "C); and P IV, 143 "C(143 "C). Benzene was used as the solvent.Commercial benzene (B.D.H. Ltd) was dried over sodium wire and distilled prior to use. The dielectric permittivity and refractive index were measured and compared with the standard values. The static dielectric permittivity was measured using a type 1312A Marconi bridge at 10 kHz. The experimental details for the measurement of dielectric permittivity and loss at 8.43, 11.3 and 35.6 GHz were as described in an earlier publication.' At each of these frequencies the attenuation constant (a)and the phase factor (p)were evaluated using the conjugate gradient function minimization technique. The entire set of data points were fitted to a numerical expression' and the best-fit set of (Y and p were obtained. For the least-squares minimization an IBM 370/155 computer was used. The temperature was controlled by circulating hot water from a thermostat (Colora K 1395) with an accuracy of kl "C in the temperature range 30-70 "C. RESULTS AND DISCUSSION The dielectric permittivity and dielectric loss for all these compounds in dilute solutions of benzene were measured at four different frequencies (10 kHz, 8.43, 11.3 and 35.6 GHz) as a function of temperature (33-70 "C) and concentration (0.01 to 0.07 weight fraction).The dielectric permittivity (F') and dielectric loss (E'') were plotted against the weight fractiom8 The slopes a' and a" at different frequencies and temperatures are listed in table 1. The a' against a" plots on the complex plane were found to be semicircular arcs with centres far below the a' axis.A typical Cole-Cole plot for P I at different temperatures is shown in fig. 1. The dielectric distribution parameter CY ', the high-frequency value aCcand the most probable relaxation times T~ were obtained at different temperatures by fitting the data to the Cole-Cole equation' and are presented in table 2. The arc plots with a non-zero CY' suggest the possibility of a multiple relaxation mechanism and the molecular structure indicate the presence of two relaxation mechanisms. Hence, double relaxation analysis was carried out using Bergmann's equations, as simplified by Bhattacharya et a1.I' The details of the least-squares minimization technique used to evaluate T~,72, C1and C2,corresponding to overall molecular orientation and intramolecular group orientation, are given by Jeyaraj and Sobhanadri.The results are given in table 3, along with the dipole-moment values calculated using the microwave and radio-frequency measurements. The thermodynamic para- meters, free energy, enthalpy and entropy of activation for overall molecular rotation were evaluated using Eyring's equation12 by assuming it to be a rate process. The calculated values are presented in table 4. The dipole moment variations with temperature and concentration are shown in fig. 2(a)and (b). The infrared spectra of P I, P I1 and P IV at 0.06 weight fraction are shown in fig, 3(a) and those of P I and P IV at 0.035 weight fraction are shown in fig. 3(b). Two bands (at 3625 and 350Gcm-l) in the i.r.spectrum of propargyl alcohol have been assigned13 to the OH unassociated species and associated species. In fig. 3(a) and (b)the infrared spectra of PI for 0.06 and 0.035 weight fractions show two bands at approximately the same wavenumbers, indicating that OH is J. G. KRISHNA AND J. SOBHANADRI TABLE1.-a' AND a'' VALUES FOR VARIOUS FREQUENCIES AND AT DIFFERENT TEMPERATURES room temperature 40 "C 50 "C 60 "C 70 "C frequency a' a" a' a" a' a'' a' a'' a' a" propargyl alcohol (0.067) static 6.33 5.83 5.55 -8.43 5.20 1.890 5.32 1.45 5.35 1.03 11.3 4.64 2.27 4.70 1.88 4.89 1.45 35.60 1.82 2.15 2.03 2.01 2.32 1.90 3-butyn-2-01 (0.07) static 6.00 -5.62 -5.31 -5.00 -4.75 -8.43 4.45 2.15 4.55 1.77 4.58 1.53 4.58 1.10 4.55 0.83 11.3 4.02 2.27 4.20 2.04 4.30 1.75 4.35 1.41 4.36 1.14 35.60 2.29 2.25 2.50 2.10 2.71 1.92 2.82 1.73 2.95 1.58 propargyl bromide (0.074) static 4.35 -3.85 -3.65 -3.25 -3.05 -8.43 2.49 0.98 2.55 0.82 2.61 0.71 2.63 0.56 2.68 0.39 11.3 2.24 0.97 2.30 0.83 2.37 0.76 2.44 0.64 2.51 0.51 35.60 1.38 0.87 1.45 0.82 1.54 0.73 1.61 0.64 1.71 0.53 2-butyn-1-01(0.063) static 9.25 -8.50 -7.75 -7.25 -6.50 -8.43 5.35 2.33 5.49 2.10 5.51 1.78 5.65 1.55 5.72 1.20 11.3 4.67 2.35 4.75 2.15 4.90 1.93 5.12 1.61 5.23 1.45 35.60 2.38 1.75 2.65 1.63 2.96 1.40 3.15 1.25 3.29 1.14 3 2 a I' 1 7 a' FIG.1.-Cole-Cole pld for propargyl alcohol at 30 (0),45 (x) and 60 "C (0). TABLE 2.-HIGH-FREQUENCY VALUE, aw, DIELECTRIC DISTRIBUTION PARA-METER, a', AND MOST PROBABLE RELAXATION TIME, T~,AT DIFFERENT TEMPERATURES compound temp./"C aW a' dPS PI 33 0.32 0.11 9.18 45 0.65 0.08 7.98 60 0.97 0.06 7.03 P I1 33 0.5 0.09 7.38 40 0.7 0.07 6.50 50 1.o 0.04 5.98 60 1.15 0.03 5.47 70 1.40 0.02 4.87 P I11 33 0.05 0.44 14.7 40 0.28 0.42 11.9 50 0.55 0.40 11.23 60 0.76 0.35 9.67 70 1.05 0.33 9.29 P IV 33 0.50 0.37 21.4 40 1.05 0.33 18.6 50 1.60 0.30 16.9 60 2.05 0.26 15.0 70 2.50 0.16 12.4 TABLE 3.-RESULTS OF DOUBLE RELAXATION ANALYSIS compound temp./"C T~/PS 72/PS c, c2 @ID (Higasi) PI 33 19.5 5.5 0.56 0.44 1.78 45 16.4 5.5 0.68 0.32 1.69 60 12.1 5.4 0.80 0.20 1.62 1.85" P I1 33 26.4 3.7 0.5 1 0.49 1.90 40 22.9 3.9 0.61 0.39 1.81 50 19.5 3.9 0.64 0.36 1.73 60 17.3 3.8 0.7 1 0.29 1.66 70 14.8 3.8 0.86 0.14 1.57 1.91" P I11 33 48.2 3.4 0.56 0.44 2.66 40 40.8 3.3 0.61 0.39 2.46 50 38.1 3.3 0.71 0.29 2.32 60 29.7 3.2 0.77 0.23 2.12 70 28.2 3.2 0.80 0.20 1.92 2.71" P IV 33 38.8 3.8 0.58 0.42 2.39 40 34.2 3.6 0.60 0.40 2.24 50 28.2 3.5 0.70 0.30 2.07 60 24.7 3.6 0.78 0.22 1.93 70 11.0 3.4 0.83 0.17 1.82 2.42" Value calculated using Guggenheim's equation.TABLE 4.-THERMODYNAMIC PARAMETERS AT DIFFERENT TEMPERATURES compound temp./"C AG*/kcal mol-' AS*/cal mol-'K-' AH*/kcal mol-' 1.66PI 33 2.93 -4.15 45 2.96 -4.09 60 3.03 -4.11 2.28P I1 33 3.08 -2.64 40 3.1 1 -2.65 50 3.13 -2.63 60 3.17 -2.67 70 3.18 -2.62 P I11 33 3.40 -3.56 2.32 40 3.47 -3.67 50 3.56 -3.84 60 3.53 -3.63 70 3.62 -3.79 P IV 33 3.31 -4.82 1.85 40 3.36 -4.83 50 3.37 -4.7 1 60 3.40 -4.66 70 3.41 -4.55 3 I A (a1 1 1 I I I 1 1 1 1 2 3 4 5 6 7 8 9 10 11 concentration/mol cm- 141 30 LO 50 60 70 temperature/'C FIG.2.--(a) Dipole-moment variation with temperature. (6) Dipole-moment variation with concentra- tion.0,P I; A,P II; 0,P III; X, P IV. 14 DIELECTRIC MEASUREMENTS ON UNSATURATED HYDROCARBONS I I I I I I I I I 1 I I I 4000 3600 3200 2800 2400 2000 1800 1600 1400 1200 1000 800 600 wavenumber/cm-' I 1 I 1 1 I I I I I I 4000 3500 3000 2500 2000 1750 1500 1250 1000 750 600 wavenumber/cm-' FIG.3.-(a) Infrared spectra (0.06weight fraction) of (-) propargyl alcohol, P I; (---* ) 3-butyn-2-01, P 11; (---) 2-butyn-1-01, P IV.(6) Infrared spectra (0.035 weight fraction) of (---) propargyl alcohol, P I; (-) 2-butyn-1-01, P IV. present both as associated and unassociated species even at such low concentrations. For PIV, two bands are present for 0.06 weight fraction whereas only one band is present for 0.035 weight fraction (3600 cm-I), indicating that OH is unassociated at this concentration. This means that the associated species of OH has slowly vanished with the increase in dilution. For P I1only one band (3600 cm-') is present J.G. KRISHNA AND J. SOBHANADRI 15 even at 0.06 weight fraction, and so OH in P I1 is unassociated for the concentration range used in the present investigation. The decrease in the values of a’ with increasing temperature indicates a significant difference in the magnitudes of the potential-energy barriers governing the rates at which the various individual relaxation processes occur. The mean relaxation time of P I11 (14.7 ps) is much larger than that of straight-chain alkyl bromide in dilute solutions of benzene14 and ally1 bromide in benzene solutions (7.7PS).~The larger T~ values for these compounds could be due to the presence of the triple bond, which reduces the intramolecular re-orientation via various C-C bonds and thus increases the rigidity of the molecule. An examination of the rovalues of all these compounds indicates that the ro value of P IV (21.4 ps) is the largest, although its molecular weight (70) is less than that of P I11 (118)and equal to that of PII.In the case of PIV, the triple bond is present at the centre of the molecule with the carbon atoms distributed equally on either side, thus enhancing the rigidity of the molecule. The distribution parameter a’ is less for P I and P I1 whereas it is larger €or P IV and P 111, which is the heaviest molecule. The analysis present in table 3 shows that T~,the overall molecular relaxation time, is much larger than T~,the intramolecular group relaxation time. Furthermore, 72 is much less sensitive to temperature compared with T’,which is characteristic of intramolecular group orientation.The group responsible for the relaxation time 72 is the OH group in P I, P I1 and P IV while it is CH2Br in P 111. For P I the 72 value (5.5 ps) is much larger than that for the other two alcohols P I1 and P IV. This is because in P I the OH group is present both as associated and unassociated species, as can be seen from the infrared spectrum. This 72 value for P I is much larger than the 72 values of OH in phenols15 and n-butanol in benzene solutions16 where the hydroxy group is a freely rotating group. The value of 5.5 ps is, however, smaller than the 72value of 8.4ps for hindered rotation of the hydroxy group as estimated by Aihara and Davies.I7 The r2value of P I11 (3.4ps) attributed to CH2Br group relaxation agrees well with the CH2Cl group relaxation (3.0-3.6 ps)I8 and that of CH2Br (2.8-3.1 ps)” in a,@-dibromoalkanes in dilute solutions of benzene and ally bromide (3.9 PS).~The weight factors corresponding to overall molecular rotation (Cl)are higher than those corresponding to group relaxation (C2),which indicates that the molecular reorientation process is the dominant factor contribu- ting to the dielectric dispersion.The T~ values of P I1 are smaller than those of P I and P IV, even though the molecular weight of P I1 is the same as that of P IV and slightly higher than that of P I. In P I1 there is a CH3 group present in the vicinity of the OH group, which reduces the association due to overcrowding.This would result in increased mobility for the OH group, which would mean a decrease in the T~ value. This is clearly seen from the r1and 72 values. values are higher than those for P I but smaller than those for PIV, where the rigidity is enhanced due to the equal distribution of carbon atoms on both the sides of the triple bond. The 72values correspond to free OH rotation, as in P IV. DIPOLE-MOMENT CALCULATIONS The dipole-moment values were calculated using Higasi’s method2’ at micro- wave frequencies as well as Guggenheim’s method2’ for radio-frequencies at room temperature. From table 3 it can be seen that the dipole-moment values calculated using Guggenheim’s equation are higher than those calculated from Higasi’s equation. The dipole-moment value of PI agrees very well with the literature 16 DIELECTRIC MEASUREMENTS ON UNSATURATED HYDROCARBONS value of 1.78 D.22A comparison of the dipole-moment values of the three alcohols reveals that the values for P I and P I1 are nearly the same (p = 1.85 and 1.91 D), while that of P IV is higher (p = 2.42 D).This may be due to the difference in the molecular structure. The dipole moment of P 111 (2.71 D) is higher than all the other liquids, owing to the highly electronegative CH2Br group. As the temperature is increased the dipole-moment values calculated using Higasi's equation decrease. It has been established that in alcohols, which exist as associated liquids in their liquid state, the dipole moment decreases with increasing temperat~re.~~ This is due to the change in the hydrogen bonding in the presence of a third oxygen atom which lowers the energy barrier for breaking the hydrogen bond.As the tem- perature is increased, the degree of dissociation of the hydrogen-bonded complexes also increases and this in turn results in a decrease in the apparent dipole moments. In the present case the presence of association causes a decrease in the dipole moment with increasing temperature. The variation is nearly linear over our experimental temperature range. The dipole-moment values for various concentrations at room temperature were calculated using Guggenheim's equation. The dipole moment decreases as the solvent concentration decreases, which means that the dipole moment decreases as the association decreases due to the decrease in concentration.Fig. 2(6) shows that the variation of dipole moment with concentration is linear, within the con- centration range used in the present investigation. The dipole moments of these liquids in solution were calculated using the conventional vector concepts.24 The dipole moment of PI obtained from these calculations is 1.91 D. This value seems to be slightly higher than the experimental and literature values. This means that there are some factors, e.g. induced dipole effects, self -association present in solution, differences in the molecular structures etc., which should be considered when calculating the dipole moment using the vector concepts. THERMODYNAMIC PARAMETERS From table 4, the values of the enthalpy (AH") are found to be lower than the free-energy (AG") values.This results in negative values for the entropy (AS")of activation, showing that the activated state of the most probable configuration involved in the dipole orientation is more ordered than the ground state. This kind of observation can be explained on the basis of co-operational orientation of The free energies and enthalpies of PI11 are higher than those of P IV which are in turn higher than those of P I, in accordance with the corresponding T~ values. In the absence of sufficient information, due to the arbitrary nature of the entropy, this suggests that the configuration involved in dipole orientation has an activated state of local order.Due to the small or negligible dependence of 72 on temperature, the thermodynamic parameters could not be calculated for the group orientation process. However, this does not mean that the internal groups do not make any resonance energy contributions. One of us (J.G.K.) thanks C.S.I.R. (India) for the financial support. We also thank Professor K. K. Balasubramanian, Department of Chemistry, for supplying the samples and for many useful discussions. A. V. Lesikar, J. Chem. Phys., 1978, 68, 1272.* L. Glaner, J. Crossley and C. P. Smyth, J. Chem. Phys., 1972, 57, 3977. J. G. KRISHNA AND J. SOBHANADRI 17 J. Crossley, Can. J. Chem., 1971, 49, 712. M. Jeyaraj and J. Sobhanadri, J. Phys. D, 1980, 13, 1925.’Handbook of Chemistry and Physics, ed.R. C. Weast (C.R.C. Press, Florida, 61st edn, 1980). R. K. Khanna and J. Sobhanadri, J. Phys. D, 1972, 5, 1453. J. Gowri Krishna, M. Jeyaraj and J. Sobhanadri, Can. J. Phys., 1981, 59, 244. W. F. Hassal, M. D. Magee, S. W. Tucker and S. Walker, Tetrahedron, 1964, 20, 2137. F. K. Fong and C. P. Smyth, J. Phys. Chem., 1963, 67, 226. 10 J. Bhattacharya, A. Hasan, S. B. Roy and G. S. Kastha, J. Phys. SOC. Jpn, 1970, 28, 204. 11 M. Jeyaraj and J. Sobhanadri, J. Chem. SOC., Faraday Trans. 2, 1980,76, 589. 12 H. Eyring, S. Glasstone and M.J. Laider, The Theory of Rate Processes (McGraw-Hill, New York, 1941). 13 R. A. Nyquist, Spectrochim. Acta, PartA, 1971, 27, 2513. 14 W. M. Heston Jr, A. D. Frankling, E. J. Hennelly and C.P. Smyth, J. Am. Chem. SOC.,1950, 72, 3443. 15 F. K. Fong and C. P. Smyth, J. Am. Chem. SOC., 1963,85, 1565. 16 G. P. Johan and C. P. Smyth, J. Am. Chem. SOC., 1969,91,6515.17 A. Aihara and M. Davies, J. Colloid Sci., 1956, 11,671. 18 S. Das Gupta and C. P. Smyth, J. Am. Chem. SOC.,1968, 90, 6318. 19 S. Chandra and J. Prakash, J. Phys. Chem., 1971,75, 2616. 20 K. Higasi, Bull. Chem. Soc. Jpn, 1966, 39, 2157. 21 E. A. Guggenheim, Trans. Faraday SOC., 1949, 45, 714. 22 S. S. Krishnamurthy and S. Soundararajan, J. Phys. Chem., 1969, 73, 4036. 23 S. D. Pradhan, S. S. Katti and S. B. Kulkarni, Ind. J. Chem., 1971: 9, 1345. 24 C. P. Smyth, Dielecfric Behatliour and Structure (McGraw-Hill, New York, 1955). ” F. H. Branin and C. P. Smyth, J. Chem. Phys., 1952, 20, 1121. (PAPER 1/1861)

ISSN:0300-9238    

DOI:10.1039/F29837900009

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1983, 79, 19-35 Image-charge Forces in Phospholipid Bilayer Systems BY BENGTJONSSON Division of Physical Chemistry 1,Chemical Centre, P.O.B. 740, S-220 07 Lund 7, Sweden AND HAKANWENNERSTROM* Division of Physical Chemistry, Arrhenius Laboratory, University of Stockholm, S-106 91 Stockholm, Sweden Received 18th January, 1982 The image-charge forces between zwitterionic phospholipid bilayers are analysed in a dielectric model for both the bilayer and the aqueous rrgion. When a second bilayer approaches surface charges are induced by the ionic groups and a repulsive force is generated. The molecular origin of the repulsion is the long-range character of the polar headgroup solvation and the second bilayer excludes some of the solvating water molecules.The effect is analysed quantitatively in a combined statistical-mechanical- electrostatic formulation. It is found that the range and the strength of the force depend in a crucial way on the zwitterionic correlations within a bilayer. For strong correlations as in a lattice the force decays exponentially with short decay lengths ( < 1A), while in the completely uncorrelated case the leading term follows a power law. The calculated magnitude of the repulsion is of the same order of magnitude as those found experimentally. We conclude that the image-charge mechanism should be considered as a possible source of the hydration force found in these systems. The explanation is appealing since its source is simply the hydroyhilicity of the polar groups.I. INTRODUCTION A number of phospholipids form bilayer structures when mixed with water. The most common phase is the lamellar liquid crystal, La, where the lipid molecules are in a liquid-like state and the bilayers are separated by water layers. At lower temperatures the hydrocarbon chains may adopt crystalline order, while the water region still has a liquid-like character. This is the gel phase, in which the lipid bilayer can be both flat and It is a common feature of both the liquid-crystalline and the gel phases that they are stable over a fairly wide concentra- tion range and the thickness of the water layer can be varied without causing a drastic change in the bilayer str~cture.~”-~ At sufficiently high water contents an additional isotropic aqueous phase appears.Since the solubility of phospholipids in water is minut2 the aqueous phase is essentially pure water, At water contents below the boundary of the two-phase region the water chemical potential is lower than that of pure water. This implies that there is an effective repulsive force between the bilayers across the water layer and that the repulsion approaches zero at the phase boundary, When a free water phase appears the bilayer is said to show maximum swelling. For pure charged phospholipids or for mixtures containing charged components there is normally a substantial swelling, which can be explained in terms of electro- static double-layer f~rces.’~-’~ For zwitterionic phospholipids such as the phos- phatidylcholines the swelling is less pronounced but an aqueous layer of substantial, 20-30 A, thickness can be obtained.For this case the origin of the repulsive force responsible for the swelling is less well understood. 19 PHOSPHOLIPID BILAYER SYSTEMS In a series of investigations Parsegian, Rand and coworkers have shown that the repulsive force, called the hydration force, is essentially exponential with a decay length of the order at 2.0-2.6 g1.7313-20The force is not sensitive to changes in the alkyl chains of the lipid^,^ it also operates in systems where a fraction of the lipids are charged13 and it is present both in the liquid-crystalline and the gel phases.20 The hydration force in the phospholipid system has also been demon- strated by Israelachvili and coworkers through direct force measurements.21 In a recent investigation22 we showed that the phase behaviour of the dipalmitoyl phosphatidylcholine-water system could be rationalized from measured hydration forces and the assumption that there are small differences in the forces in the different phases.Hydration forces have also been found experimentally in the interaction between mica sheets across an aqueous layer ~ystem.~~'~~ The existence of the hydration force is thus clearly documented and there also exists a substantial body of data on the quantitative properties of the force. In contrast, the theoretical understanding is still limited, although several suggestions have been made concerning the origin of the hydration force.At an early stage Par~egian~~suggested that in a zwitterionic system a conventions1 double-layer force is also operating. However, it was realized that there are impoi iant qualitative differences between ionic systems, where an important contribution to the free energy comes from the entropy associated with the ion distribution, and the zwitterionic case, where the charges are linked covalently. This led Colbow and Jones26 to consider the direct electrostatic interaction between two surf aces with a distribution of dipoles. With a smeared-out uniform dipolar density the interaction is zero, but for dipoles of a molecular nature correlations between dipoles on the two surfaces leads to an interaction of a second-order nature.Owing to the second-order character of this interaction it is always attractive and weak and it cannot explain the repulsive force. This insight led Marcelja and Radi~~~ to consider a force associated with the molecular nature of the solvent (water). The basic idea is that the bilayer surface orients the water molecules in direct contact with the surface. This orientation is then propagated, in a damped way, out from the surface through successive layer interactions. Using a Landau-type free-energy expansion Marcelja and Radic showed that an exponential repulsive force could be derived from the model. Recently Marcelja and c~workers~*~'~~ have presented a molecular formulation introducing a spatially dependent electric polarization.This model is based on an analysis of the distribution of defects in an ice-like structure and the decay constant is found to depend on the defect density. Helfrich3' has developed a quite different model for the repulsive bilayer interaction based on short-range interactions and a description of the elastic motions of the bilayers. The amplitude of the elastic motions should decrease as two bilayers approach due to the short- range repulsion. This gives an excluded-volume contribution to the free energy resulting in an effective repulsive force, which is found to vary as the inverse cube of the bilayer separation. Yet another possibility was considered by Levine and Bell,32 who specifically considered the non-specific adsorption of electrolyte ions due to the presence of a dipolar potential.The adsorbed ion then gives rise to a repulsive double-layer force. In our opinion none of these explanations of the repulsive force is at the present stage quite satisfactory and we were therefore motivated to investigate alternative possibilities. In the present paper we describe yet another mechanism for the repulsive interaction between bilayers. The model is developed by first considering the chemical knowledge of the behaviour of zwitterionic amphiphiles in both micellar and liquid-crystalline systems. Attention is focused on the nature of the interaction B. JONSSON AND H. WENNERSTR~M between water and ionic groups, which is mainly of an electrostatic origin.In a bilayer system the hydration will be more complete the greater the thickness of the aqueous layer, leading to an effective repulsive force. The model is formulated in quantitative terms using the concept of image charges. For a few illustrative examples the hydration force is calculated explicitly and it is shown that the effect can be large enough to account for the observed repulsive interaction. 2. SOLVATION OF POLAR GROUPS Phosphatidylcholines (PC)belong to the general group of amphiphilic molecules with the characteristic properties of an apolar (hydrocarbon) chain and a strongly polar headgroup. The ambivalence of the amphiphiles in their interactions with water leads to self-association in an aqueous medium such that the aggregates are finite in at least one dimension.33 Thus, depending on the particular amphiphile and on the concentration, micellar solutions or liquid-crystalline phases are Phosphatid lcholines have two hydrocarbon chains and geometrical considerations show3373' that a bilayer structure is the most favoured.For lysolecithin with the same headgroup but with only one apolar chain, micellar aggregates are stable over an appreciable concentration range.37 This indicates that the zwitterionic choline group has a strong affinity to an aqueous medium and not only to one or a few directly bound water molecules. It is also clear from the analogy with other polar groups38 that the affinity to water is mainly of an electro- static origin. For a free ion in solution the major part of the hydration energy is due to interactions with water molecules in the first hydration shell.However, the hydra- tion interaction has a long-range tail. This is most clearly seen in the dielectric approximation of aqueous medium. In this model the contribution to the free energy of hydration of a spherical shell decays with distance as r-2. When ions are arranged in some array of charges where the total system is electroneutral the range of the hydration interaction is reduced but it can still give rise to appreciable long-range effects, as will be demonstrated in this paper. In a lamellar PC system the water in the aqueous region will provide the solvation for the zwitterionic groups. The thickness of the aqueous layer varies with the water content, leading to changes in the hydration interaction. A qualitative description of this effect is obtained by considering a single zwitterion fixed at a bilayer surface.Outside the surface there is an aqueous region and then a new interface towards a hydrocarbon-like medium (see fig. 1). It is assumed that the long-range part of the electrostatic interaction can be described in the dielectric approximation. The high dielectric permittivity of water implies that there are strong dielectric discontinuities at the two interfaces. The zwitterion induces polarization charges at the interfaces with the net result that there is a repulsive force across the aqueous region. Image charges can be used to visualize the force arising from the dielectric discontinuities, as illustrated in fig.1. Using the image- charge concept one should remember that the physical effect is the induced polarization charges at the interfaces, while the image charge is just a device to construct an equivalent system where the net interaction is the same. In the PC system it is clearly not sufficient to consider an isolated zwitterion. The induced polarization charges, or the image charges, interact with all zwitterions in the system. This leads to substantial complications in the quantitative analysis, but it will be shown in subsequent sections of this paper that such an analysis can be performed once certain approximations are introduced. P H 0S P H 0LIPID B I LAY E R SYSTEMS HC H-0 HC FIG.1.-Schematic representation of two bilayer surfaces with zwitterionic groups facing each other. One of the virtual image charges induced in the hydrocarbon region is also shown. E~ and E~ are the relative dielectric permittivities of the hydrocarbon and the aqueous regions, respectively. 3. INTRA- AND INTER-LAMELLAR INTERACTIONS Electrostatic interactions in lamellar systems show some non-trivial features. Considerable help in appreciating the subtleties is obtained through a formal statistical-mechanical analysis of the problem. We write the total energy as a sum of contributions from short-range interactions, ESR, and from the electrostatic interaction, Eel, E = ESR + Eel. (1) By assumption, the short-range interaction is only acting between lipids in the same bilayer, between lipids and water in direct contact and between water molecules in the bulk.In the following the effects of the short-range forces will be described through parameters and the interest is instead focused on the electrostatic interac- tion, the long-range part of which should be describable in a dielectric model for the medium. In this case only the charges on the zwitterion need to be considered explicit1y. The electrostatic energy can be written as where Fl> is the electrostatic potential generated at F, by a charge qa in position Fs, and p:; (Fs,7,) is the two-particle distribution function for particles of species a and /3 in lamellae s and t, respectively. Eqn (2) can be simplified by separating interactions between charges on different lamellae from the intralamellar ones.With s # t it is a reasonable assumption that the correlation between charges on different lamellae is weak and B. JONSSON AND H. WENNERSTROM where the last equality follows from the translational symmetry in the lateral direction. Since there is an interaction between charges on different lamellae the first equality in eqn (3) can be seen as describing the first term in a series expansion. The second term will always give an attraction, and this is the interaction that was explicitly considered by Colbow and Jones.29 In this paper we conbentrate on the first-order effects and investigate the electrostatic interactions that occur even in the absence of correlations between different lamellae.Since there is elec- troneutrality at the bilayer surface the approximation in eqn (3) implies a charge distribution of a smeared-out dipolar layer. In planar symmetry the interaction between two such layers is zero. It follows from Gauss' law that the field outside the charge distribution is zero and there is thus no interaction with a dipole. This conclusion also holds in the presence of dielectric discontinuities as long as the planar symmetry is retained. For the case when Fs and Ft are on the same lamella the interaction energy can be divided into two parts; one describing the direct interaction, Eintra,which is ,present also in the absence of other lamellae, and one, smaller, contribution, Elnfer,that depends on the bilayer separation such that it is zero at infinite bilayer separations, 2,. For the electrostatic potential the separation can be written where the parametric dependence of Qinter on the bilayer separation zw has been indicated. The interlamellar interaction is E',?te'(z,) = C cD:ter [F5, Yr(zm)]p$(f5,Y,) dF: dF7 (5)ad il assuming the validity of eqn (3).Counted per degree of freedom this interaction is small relative to kT and one can neglect the entropic contributions to the free energy; the effective force is then obtained directly as the derivative of EZtr'. Furthermore, the dominant interactions in determining pfi (Fs, Ft) are the short- range interactions ESRand the direct electrostatic interaction EL:tra,while E~"' has a minor influence.It is known that the area per polar group does change with bilayer ~eparation,'~~ which implies that E:"' has an influence on P'~'.However, in order to keep the description simple we will neglect the coupling betwzen E:ter and P'~'.At large separations it should be negligible and it a pears that even at shorter distances the effect is only of secondary importance. 16,2? The problem of determining the interlamellar interaction has now been reduced to an evaluation of the integral in eqn (5). In the following the two-particle distribution function will be considered as a parameter whose qualitative properties might be known while the quantitative ones are not. The potential Qzter,on the other hand, can be determined by solving the electrostatic equations with the proper boundary conditions.Before turning our attention to the electrostatic problem, we note that pfi(Fs, Y,) have a &function peak at f, = ?t when CY = p. Two types of contributions to EZtercan now be recognized E',;ter= Cqp @;te'( Yy)pg' (7,) d?: P PHOSPHOLIPID BILAYER SYSTEMS where the first term on the right-hand side represents the interaction between a charge and its own images, while the second term represents the interaction between the images and all other charges in the same lamellae. For the case of a zwitterion, p$ contains one contribution from the other charge in the zwitterionic pair and in addition contributions from all the other zwitterions in the system.Since for a uniform zwitterion distribution there is no interaction, the contribution from zwit- terions other than the source can be viewed as due to a diffuse zwitterion hole around the source. There can thus be substantial cancellations between the two terms in eqn (6). The weaker the correlations between two zwitterions the more diffuse is the hole and the weaker is the cancellation effect. We thus reach the important conclusion that the repulsive interaction between the bilayers due to the image effect depends crucially on the correlation between the charges within the bilayer. The weaker the correlation the larger the interaction energy. 4.THE ELECTROSTATIC POTENTIAL Consider a lamellar system with a succession of layers and with a charge, q, fixed in layer i = 1at (z=a, x2+y2=0), as illustrated in fig.2. The system possesses cylindrical symmetry and the electrostatic potential at a point [r = (x2+Y~)”~,z] can be expressed in terms of an integral containing a zero-order Bessel function J~(X)39 (7) This equation is valid in all regions i, while the values of A,(k) and Bi(k) are determined by the boundary conditions. The potential and the dielectric displace- ment are continuous at the interface (z=zi)between regions i and i +1so that B. JONSSON AND H. WENNERSTROM X FIG.2.-Representation of the coordinate system used in the calculation of the electrostatic potential. i?h and z, are the thicknesses of the bilayer and the aqueous regions. The source charge is at z = a ; x, y = 0.Since the potential should remain finite a comparison with eqn (7) reveals that Ai(min) and Bi(max) must vanish. Here i(min) and i(max) refer to the last regions at negative and positive values of z,respectively. From the similarity of the recursion relations of eqn (12) and (13) it then follows that if there are the same number of bilayers on both sides of i = 0. In eqn (14) a quantity Y has been defined to simplify the notation and E~ and E2 are the dielectric permittivities of the hydrocarbon and water layers, respectively. An explicit solution for the potential will only be needed in the region where there are charges in the lamellae. Below we will assume that the charges are situated at the dielectric discontinuity and at the low E~ side of this boundary.With a bilayer thickness of tb we have for the region -tb<22 <zb PHOSPHOLIPID BILAYER SYSTEMS where the constant has been introduced. Eqn (15) gives the potential for an isolated bilayer. With several bilayers present the change in the potential is dnter' f low= I___ [sinh (ka)sinh (kz)2(Xu-l)-m-l47T€1&0m=l + cosh (ka) cosh (kz)2(1+XJr"--'JX,"YmJo(kr)dk (17) where Y is as yet undetermined. For a lamellar liquid-crystalline system the set of eqn (12) and (13) become infinite but an approximate solution can be obtained by considering a few bilayers only. In the case of four interfaces Y is given by where zwis the thickness of the water layer. Four interfaces implies that, from a dielectric point of view, two bulk phases of low dielectric constant are surrounding the central bilayer.The repulsion calculated from this configuration should provide an upper limit for that obtained with a multi-bilayer. Six interfaces, on the other hand, represent three bilayers surrounded by bulk water and in this case one would obtain a lower limit for the interaction, For six interfaces a2 Ys= Y4 1 x? (1-Y4)". (19) n =O For a larger number of interfaces the equation for Y will be still more complex but the lowest term in a power series of Xo is always Y4.This implies that, when one is only considering the first two terms (rn = 1,2) in the summation of eqn (17), it is sufficient to use Y4for Y. It follows from eqn (16) that Xo<<1 unless k is small enough to make kzb< 1.It is only for these values of k that the higher-order (rn > 1) terms in eqn (17) contribute significantly to the integral. However, the integral of eqn (15) determining the energy contribution to the potential from small values of k will be practically constant over a sufficiently large region so the electroneutrality ensures no net contribution to the energy. It is consequently a good approximation to retain only the rn = 1 term of eqn (17) when calculating the interaction energy and where eqn (16) and a series expansion of eqn (18) have been used. A further approximation can be introduced by realizing that the charges are located in the vicinity of the bilayer boundary so that -2b/2 << a, z < 2b/2. Recalling that we are mainly interested in the value of the integral for k >zb, the exponential terms in B.JONSSON AND H. WENNERSTROM FIG.3.-Hexagonal perfect lattice of zwitterions. eqn (20) are dominated by exp [-k(zb-a -z)] and the equation is simplified to 2p+l @:'"'[0, a; r, z(z,)]=-This is the bilayer-separation-dependent part of the potential at position r, z generated by a charge qa in the position r = (x2+y 2)1'2 = 0, z = a. This expression will be used in the subsequent calculations of the interaction energy. In calculating the energy EZterof eqn (6) we will depend on an assumed model for the correlation function p'?)(Tr, 7[). One conceptually important limiting case is the one with perfect correlation so that the zwitterions form a lattice, as illustrated in fig.3. In this particular case the formalism outlined above is cumbersome to use since ~(~'(f~,ft) does not decay to p(l)(Ts)p(l)(Tr) at large distances. The symmetry in the lateral direction can, on the other hand, be used to advantage by approaching the problem of determining the potential from a slightly different angle. The total potential is equal at equivalent positions in each, hexagonal, unit cell and the field, in the perpendicular direction, is zero at the cell boundary. This boundary condition is mathematically simpler to apply in the nearly equivalent problem that is obtained by approximating the hexagonal geometry by cylindrical geometry. When the condition = 0 (Rcylinder radius) is applied in eqn (7) one obtains where y, is the nth zero of the first-order Bessel function J1(x).In applying the boundary conditions of eqn (8)and (9) one finds that it is the term with n = 1 that PHOSPHOLIPID BILAYER SYSTEMS dominates in eqn (22),except at very small bilayer separations.Using this approxi- mation The first zero of the first-order Bessel function is yl =3.832 and in this point is Jo(yl) = -0.4028. Having derived eqn (21) and (23) for the electrostatic potential we now turn to the calculation of the interaction energy. 5. THE INTERACTION ENERGY To ap ly eqn (6) in calculating the interaction energy the distribution functions p(')and pRi have to be specified. The simplest possibility is to consider the hexagonal lattice of fig.3. For a zwitterion having two unit charges a distance A apart but at the same coordinate r in the lateral direction a direct application of eqn (23) gives where ad is the number of dipoles and one of the charges has been placed at z =a. The interaction energy depends exponentially on the water-layer thickness zw, with a decay length of R/2y1. With a typical area per molecule of 65 A2the decay length is <1A and thus a factor two to three shorter than what was experimentally observed. Note that for a lattice the approximation of eqn (3) is no longer valid. Due to the perfect coupling between positions of zwitterions in the same lamellae a correlation can be generated between the position in two different lamellae without loss of entropy.In the electrostatic model an attraction would then arise due to the direct zwitterion-zwitterion interaction. The regular lattice still rep- resents an interesting limiting case for the image-charge effects. In a liquid-crystalline state the model with a regular lattice of dipoles is not realistic. Even in the gel phase where the hydrocarbon chains are ordered the headgroups are still in the aqueous region and show considerable conformational disorder, as seen from deuterium n.m.r. These measurements also show that the zwitterionic headgroup of the phosphatidylcholines is oriented parallel to the bilayer surface rather than perpendicular to it. In order to model this behaviour, at last qualitatively, we have considered the following model for the conformation of the headgroups and for their correlation.(i) The zwitterions are arranged so that there is a charge -e at Fs and a charge e at r', +A, where the length A is the same for all zwitterions. (ii) The correlation between the negative charges of different zwitterions is described by a pair distribution function g(r)[r =(x2+Y~)~'~]. (iii) The angle 8 between the normal to the bilayer and the direction of & is the same for all dipoles, so that both the negative charges and the positive charges form one plane each. (iv) It is assumed that there is no correlation between two angles 41and 42 describing the azimuthal orientation of two zwitterions. The potential from the pair of charges of the zwitterion is obtained from eqn (21) by the principle of superposition 2ptl 30 e~2 f (FI-E~) J: exp {-k [2(p +l)tw@;ter = +zt,-a -to]}T&o(&2+&1)2p=o &2+&1 x{-JO(kr)+exp (kAcos B)Jo[k(r2+A2 sin20 -2rA sin 0 cos 4)1'2]}dk.(25) B. JONSSON AND H. WENNERSTROM The second Bessel function can be rewritten using the addition theorem for Bessel functions4' to3 Jo[k(r2+A2sin26-2rAsin8 COS~~"~]=1 J,(kr)J,,(Asin 8)exp(in#). n=-a (26) Due to assumption (iv) in the model only the value of averaged over all q5 is needed in the calculation. In this case only the term with n = 0 survives in eqn (26) and the averaged value of the potential is x lomexp -k [2 + 1 zwr+ zb -a-zI> x [Jo(kAsin 8)exp (kA cos 8)---l]Jo(kr)dk (27) This expression is now inserted into eqn (6) to yield the interaction energy +[Jo(kAsin 8)exp (kA cos 8)-11' where nd/A is the zwitterion density.Here the three first terms in the curly bracket in the integral are due to the interaction between the zwitterion and its own image, while the last term is due to the interaction between the image and the other zwit terions. To evaluate the interaction energy of eqn (28) the pair-distribution function has to be specified. At present the form of this function is unknown. Apart from the rigid lattice already treated two different forms of g(r) will be considered to illustrate the qualitative features of the interaction. If the rigid lattice is one extreme the other is represented by g(r) = 1, which implies no correlations.In this case there is no net interaction between the image of one zwitterion and the othcr zwitterions since they appear as a smeared-out dipolar density. This can also be shown explicitly from the properties of Bessel functions. The remaining integral over k containing exponential and Bessel functions is readily performed and (2% where L, stands for L, = 2(p + l)z,+zb-2a-A cos 8. At larger separations when z, > A, p = 1gives the dominating term in the summation and the interaction is proportional to (2~,)-~.This result could of course be PHOSPHOLIPID BILAYER SYSTEMS obtained simply by considering the interaction between a dipole and its own image in a direct calculation. In the real bilayer the separation g(r) is intermediate between a rigid lattice and the completely uncorrelated situation where g (r)= 1.We expect short-range correlations which are then damped at larger separations similar to what is found for the three-dimensional g(r) in a liquid. It is mathematically convenient to write g(r) as an expansion in first-order Bessel functions which simplifies the evaluation of the integral in eqn (28). The constants a, and r, > 0 can be chosen so that the expected features of g(r) are reproduced. From normalisation one obtains co nd/Alom27~r[l -g(r)] dr = 1 -rS C a,r, =A/(27~n,). (31),=l At the origin g(r)should vanish and 00 n=l The integral over r in eqn (28) uses eqn (30) 00 2nrg(r)Jo(kr)dr = 2.rrrJ0(kr) dr -277 C a, lo*Jl(r/r,)JO(kr)dr. (33)lom lo* ,=I The first integral gives no contribution to the interaction energy for the same reason as the g(r)= 1 case.The second term contains a Weber-Schafheitlin integral4' which equals r, if k < l/r, while it is zero otherwise. In evaluating the interaction energy only the case 8 = 0 is considered for simplicity. When 8 # 0 a more involved expression is obtained but the main features are the same. With 8 = 0 a3 x a,r, exp(-[2(9 +l)zw+zb--2a]/rn) (34),=1 The dominant term when the thickness of the aqueous layer zwexceeds the distance A between the zwitterionic charges is Note that the leading term is essentially exponential and that the introduction of a partial correlation has changed the distance dependence of the interaction from a power law to an exponential.B. JONSSON AND H. WENNERSTROM 5 10 15 20 rlA FIG. 4.-Graphic representation of the pair-distribution function gfr) for zwitterions in the bilayer given in eqn (36). The first maximum in g(r)occurs at the average zwitterion distance of 8 A. To illustrate the use of the expansion of g(r) in Bessel functions we have constructed a two-term expansion of g(r) g(r)= 1-[3.08Jl(r/1.7)+0.91Jl(r/5.7)]/r (36) with distances in A. This g(r)is shown in fig. 4 and the constants a, and r, have been chosen so that the first maximum in g(r)occurs at the distance between two zwitterions in an hexagonal lattice with an area of 65 A per molecule. 6. INTERLAMELLAR FORCES The repulsive interlamellar interaction gives rise to an osmotic pressure d(E:'"'/A)7T= -2 (37)dzw where the factor two arises from the fact that there is one contribution to EZfer from each lamella.It is straightforward to evaluate the derivative from the expressions for EZterin eqn (24),(29) and (34). To arrive at numerical values the following parameters are used A/nd= 65 Hi2 (R= 4.55A) A=5A &1=2 ~2=78.4 zb--2a-2A cos 8 = 0. The last equation implies that the positive charge of the zwitterion is situated on the surf ace of the dielectric discontinuity. At longer bilayer separations, i.e. when P H0S P €10I, I PI D B I LAYER S YSTEMS zw>A, it is sufficient to retain the leading terms in the expression for 7~ (in N mW3) and for (9 gm = 1 (ii) hexagonal lattice 7r 1x lo9exp (- zW/O.6) (40) (iii) g(r)as in eqn (36) 7~ =8 x 107zi1 exp ( -zW/2.92j.(41) In all three equations zw are given in Angstroms. Eqn (39)-(41) clearly show that the mere long range the correlations within the bilayers the more short range is the repulsive force. The reason that g(r)of eqn (36) gives a near exponential force is that g(r) approaches its limiting value of unity rather slowly. For g(r)= 1 the force is of a dipole-dipole type. With g(r) of a square well type the dominating term at large separations is an sctopole-dipole interaction and the force decays as zw ,illustrating the sensitivity of T towards the choice of g(r). In the experimental determinations of the repulsive force the osmotic pressure, or an equivalent quantity, is obtained as a function of the water-layer thickness, d,.This quantity is calculated from the repeat distance measured by X-ray diffrac- tion and from the composition, assuming incompressibility and a total division between polar and agolar regions. In reality the dielectric constant will vary as one moves from the polar bilayer interior out into the aqueous region. It is not clear where it is appropriate to place the dielectric discontinuity. It is generally found that only for one or two layers of water molecules are the molecular properties appreciably different from those in bulk It has been found from structural models44 that approximately eleven water molecules interact directly with a phos- phatidylcholine.If the dielectric discontinuity is placed outside this first layer one should add 2~(30.11/65)=10Ato the values of d, to obtain z,. This should only be considered as a first approximation. Using this choice of the relation between zwand d, the calculated osmotic pressures are compared with the experi- mental values for egg lecithinI4 (see fig. 5). The case with no correlations gives consistently larger values than observed while in the rigid lattice the osmotic pressure decays rapidly with distance, The good agreement between the value calculated using g(r) as in eqn (30) and the experimental values is partly achieved through parameter fitting. The calculations show that the image-charge effect might be large enough to accour,t for experimefital values.7. DISCUSSION The calculations in the previous sections clearly show that the long-range part of the hydration of the zwitterion does give rise to an appreciable force. Although objections can be raised against the dielectric approximation for the solvent the indications are that it does apply for electrostatic effects over distance >10 A, at least. This conclusion is based on the success of the Poisson-Boltzmann 2745-4' and on direct dielectric rneasurernent~.~~ apprOachl~,l It follows that the image-charge effect ought to be included in a complete theory of interbilayer forces in phospholipid systems. It is more problematic to assess if the image-charge effect R. JONSSON AND H.WENNERSTROM 7 z 4.-25 3.v w -4 2. 3 2 \1. 10 20 30 dwlA FIG.5.-Calculated osmotic pressure Tfrom image-charge effects for four different cases: (1)hexagonal lattice, (2) pair-distribution function as in eqn (36)and fig. 4,(3)no intrabilayer zwitterions correlations, zwitterion parallel to the biiayer plane, (4) as in (3)but with the zwitterion perpendicular to the bilayer. The dots represent experimental values of LeNeveu et al. from ref. (14). is dominating relative to other contributions. The calculations presented above contain uncertainties in the physical model that give rise to an uncertainty in the calculated pressure of an order of magnitude. Probably the strongest argument in favour of the image-charge effect is that its basis is found in the typical features of amphiphilic molecules. At short bilayer separations with only a few (5-10) water molecules per lipid the continuum picture should break down and one has to consider the molecular nature of the problem.However, experience with ionic systems12 shows that the breakdown of the continuum model is not dramatic. This is consistent with the very regular variation in the osmotic pressure even at very low water contents.20 At these short separations the correlations between zwitterions on opposing bilayers can no longer be neglected and some of the approximations made in section 3 are no longer appropriate. To find a proper treatment of the bilayer system with a few water molecules per lipid one will in the end have to use computer simulation techniques.To conclude, we point to some qualitative experimental observations that can be rationalized with the present model for the hydration force. As discussed in ref. (22) the hydration force is stronger in the liquid-crystalline phase of dipalmitoyl phosphatidylcholine than in the gel phases. This is consistent with a weaker correlation in the liquid crystal due to both translational and orientational disorder of the polar headgroups. An additional factor is the larger area per headgroup. With a solute like cholesterol in the phospholipid bilayer the correlation in g(r) should decrease and the average distance between the zwitterions should increase. Both these factors should give rise to a longer range of the hydration interaction while it should be weaker at shorter separations. Concerning hydration forces in other types of systems such as the mica-water system studied by Israelachvili and Pashley, it would require a detailed study to assess if the image-charge effects are important in this case also.Mica has a low relative permittivity and the electrostatic interactions are certainly influenced by 34 PHOSPHOLIPID BILAYER SYSTEMS this fact, and in a quantitative treatment one would have to consider dielectric effects not only on the field charges but also on the free ions in the aqueous region. Such a theory remains to be developed in quantitative terms. The bilayer repulsive forces are not particularly sensitive to the presence of salt,20 which could be taken as an indication that a non-electrostatic explanation should be involved.However, the decay length of the repulsive force is short and it would require high salt concentrations to obtain Debye screening lengths of the same magnitude. The role of the electrolyte in screening the image forces will be discussed in a separate paper. V. Luzzati and A. Tardieu, Annu. Rev. Phys. Chem., 1974, 25, 79. * R. P. Rand, D. Chapman and K. Larsson, Biophys. J., 1975, 15, 1117. M. J. Janiak, D. M. Small and G. G. Shipley, Biochemistry, 1976, 15, 4575. J. Stamatoff, B. Feuer, H. Guggenheim, G. Tellez and T. Yamane, Biophys. J., 1981, 33, 141a. D. Chapman, Biol. Membr., 1968, 1, 125. Y. Inoko and T. Mitsui, J. Phys. SOC.Jpn, 1978, 44, 1918. ' L.J. Lis, M. McAlister, N. Fuller, R. P. Rand and V. A. Parsegian, Biophys. J., 1982, 37, 657. * J. Ulmius, H. Wennerstrom, G. Lindblom and G. Arvidson, Biochemistry, 1977,16, 5742. L. Powers and P. S. Pershan, Biophys. J., 1977, 20, 137. 10 E. J. W. Verwey and J. Th. G. Overbeek, The Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948).11 V. A. Parsegian, Trans. Faraday SOC.,1966,62, 848. 12 B. Jonsson and H. Wennerstrom, J. Colloid Interface Sci., 1981, 80, 482. 13 A. C. Cowley, N. L. Fuller, R. P. Rand and V. A. Parsegian, Biochemistry, 1978, 17, 3163. 14 D. M. LeNeveu, R. P. Rand and V. A. Parsegian, Nature (London), 1976,259,601.15 D. M. LeNeveu, R. P. Rand, V. A. Parsegian and D.Gingell, Biophys. J., 1977,18, 209. 16 V. A. Parsegian, N. Fuller and R. P. Rand, Proc. Natl Acad. Sci. USA, 1979,76, 2750. 17 R. P. Rand, V. A. Parsegian, J. A. C. Henry, L. J. Lis and M. McAlister, Can. J. Biochem., 1980, 58, 959. 18 L. J. Lis, V. A. Parsegian and R. P. Rand, Biochemistry, 1981, 20, 1761. 19 L. J. Lis, W. T. Lis, V. A. Parsegian and R. P. Rand, Biochemistry, 1981, 20, 1771. 20 R. P. Rand, Annu. Rev. Biophys. Bioeng., 1981, 10, 277. 21 R. Horn and J. Israelachvili, to be published. 22 L. Guldbrand, 5. Jonsson and H. Wennerstrom, J. Colloid Interface Sci., 1982, in press. 23 J. N. Israelachvili, Faraday Discuss. Chem. SOC.,1978,65, 20. 24 J. N. Israelachvili and R. M. Pashley, in Proceedings from Wafer in Biological Systems, ed.F. Franks, 1982, in press.25 V. A. Parsegian, Science, 1967,156, 939. 26 K. Colbow and B. L. Jones, Biochim. Biophys. Acta, 1974, 345, 91. 27 S. Marcelja and N. Radic, Chem. Phys. Lett., 1967, 42, 129. 28 S. Marcelja, D. J. Mitchell, B. W. Ninham and M. J. Sculley, J. Chem. SOC.,Faraday Trans. 2, 1977, 73, 630. 29 D. W. R. Gruen, S. Marcelja and B. A. Pailthorpe, Chem. Phys. Lett., 1981, 82, 315. 30 D. W. R. Gruen and S. Marcelja, J. Chem. SOC.,Faraday Trans. 2, 1983, 79, 211 and 225. 31 W. Helfrich, 2. Naturforsch., Teil A, 1978, 33, 305. 32 P. Levine and G. Bell, J. Colloid Interface Sci., 1980, 74, 530. 33 C. Tanford, The Hydrophobic Efect (Wiley, New York, 2nd edn, 1980).34 H. Wennerstrom and B. Lindman, Phys.Rep. 1979, 52, 1. 35 G. J. T. Tiddy, Phys. Rep., 1980, 57, 1. 36 J. N. Israelachvili, D. J. Mitchell and B. N. Ninham, J. Chem. SOC.,Furuday Trans. 2, 1976, 72, 1525. 37 F. Reiss-Husson, J. Mol. Biol., 1967, 25, 3631. 38 R. G. Laughlin, Adv. Liq. Cryst., 1978, 3, 41. 39 J. P. Jackson Classical Electrodynamics (Wiley, New York, 1975).40 J. Seelig, Q. Rev. Biophys., 1977, 10, 353. 41 J. H. Davis, Biophys. J., 1979, 27, 339. 42 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1956). B. JONSSON AND H. WENNERSTROM 35 43 B. Halle and H. Wennerstrorn, J. Chem. Phys., 1981, 75, 1928. 44 E. Forslind and R. Kjellander, J. Theor. Biol., 1975, 51, 97. 45 G. S. Manning, Q. Rev. Biophys., 1978,11, 181. 46 J. N. Israelachvili, Nuture (London),1976, 262, 775. 41 G. Gunnarsson, B. Jonnson and H. Wennerstrom, J. Phys. Chem., 1980, 84, 3114. 48 Y. Hamnerius, I. Lundstrom, L. E. Paulsson, K. Fontell and H. Wennerstrorn, Chem. Phys. Lipids, 1978, 22, 135. (PAPER 2/104)

ISSN:0300-9238    

DOI:10.1039/F29837900019

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1983, 79, 37-40 Localization of Molecular Orbitals BY T. J. TSENG? AND M. A. WHITEHEAD* Department of Chemistry, McGill University, Montreal, PQ, Canada H3A 2K6 Received 5 th April, 1982 The degree of localization is the same, irrespective of the localization chosen, in CNDO methods. There has been a reawakening of interest in localized orbitals for chemical and the use of their net charges and gross atomic populations to interpret the redistribution of atomic charges resulting from chemical bonding. This paper shows that there is no unique localization that can be obtained, and consequently analysis of the properties of a localized structure is of little significance in CNDO and similar methods. DEGREE OF LOCALIZATION For molecular water, with bond angle 8 = 104.5" and bond length R (OH)= 0.9572 A, where the y axis bisects the angle 6, x is perpendicular to y in the LHOH plane and z is perpendicular to the molecule, the degree of localization of a set of doubly occupied orbitals is3 CP can be canonical molecular orbitals (CMO) or localized molecular orbitals (LMO).The LMO for H20using the CNDO/BW scheme4 are I1 = -0.784 3841 +0.620 2944 /2=-0.355 2341-0.577 9243-0.449 2044-0.581 4045-0.005 1846 l3 = 0.355 23~$~-0.577 9243 +0.449 2044 +0.005 18~&+0.581 40& 14 = 42 where 41 = 2s(O), 42 = 2p,(O), 43 = 2p,(O), 44 = 2p,(O), 45 = ls(H1) and 46 = 1s 032). Thus eqn (2)gives two equivalent bonding orbitals and two inequivalent lone-pair orbitals. The CMO ar:-+iven in table 1.However, some reports gave two equivalent lone-pair orbitals, these being the ab initio SCF calculations of Liang7 and the semi-empirical calculations of Hunt et al.? which show the total energy of the ground state of H20 to be tower by 0.0884 eV for equivalent lone-pair orbitals: the energy lowering was increased when the exchange energy was excluded.' f Permanent address: Chung Yuan Christian University, Chung Li, Taiwan. 37 LOCALIZATION OF MOLECULAR ORBITALS TABLE1.-CNDO/BW CMO AND LMO OF H,O[R(OH) =0.9572 A, t) =104.5’1 2s(0) 2Pz (0) 2Px(0) 2P, (0) 15wd 1s (H,) CMO 2al 0.905 69 0.0 0.0 0.028 35 0.299 10 0.299 10 1b2 0.0 0.0 0.817 30 0.0 0.407 45 -0.407 45 3a1 -0.217 57 0.0 0.0 0.887 42 0.287 36 0.287 36 161 0.0 1.0 0.0 0.0 0.0 0.0 LMO 11 -0.784 38 0.0 0.0 0.62029 0.0 0.0 12 -0.355 23 0.0 -0.577 92 -0.449 20 -0.581 40 -0.005 18 4 0.355 23 0.0 -0.577 92 0.449 20 0.005 18 0.581 40 4 0.0 1.o 0.0 0.0 0.0 0.0 1; -0.554 64 -0.707 11 0.0 0.438 61 0.0 0.0 1; -0.355 23 0.0 -0.577 92 -0.449 20 -0.581 40 -0.005 18 1; 0.355 23 0.0 -0.577 92 0.449 20 0.005 18 0.581 40 I& -0.554 64 0.707 11 0.0 0.438 61 0.0 0.0 In the present work no energy difference can occur because the LMO are unitary transformations of the CMO.9-12 When a linear combination is made of ll and l4 from eqn (2) (3;!1 while 1; =12 and 1; =13, a new set of LMO is generated, and the determinantal function is normalized and unchanged.The two equivalent lone-pair orbitals 1; and 1: are 11; =~2(-0.784 3841-43+0.620 2944) with 1; =12 and 1; =13, the equivalent bonding orbitals. Using eqn (1) and the coulomb integrals from the CNDO/BW calculations, table 1, the D values can be calculated for both sets of LMO for H20in terms of the canonical molecular-orbital basis T. J. TSENG ANT) M. A. WHITEHEAD TABLE2.-CNDO/BW COULOMB INTEGRALS (IN eV, IN CMO FOR GROIJND-STATE H20 [R (OH)= 0.9572 A, 8 = 104.5"] 0 13.6250 9.9963 9.9963 HI 9.9963 12.8480 7.6443 H2 9.9963 7.6443 12.8480 where Rijkl is the two-electron integral r 1 The CNDO/BW approximation reduces eqn (6) to where N is the number of LMO and M is thc number of CMO.D(I') can be similarly expressed: D(I) =D(l') = 38.4282. Thus, within the CNDO/BW scheme the degree of localization calculated with the orbitals from eqn (5) is the same as that using the orbitals in eqn (2). Both the inequivalent lone pair, sp2 and pz, and the equivalent lone pair, sp', are equally localized, CONCLUSIONS Within the CNDO scheme the degree of delocalization, the energies of the LMO and the charges will be independent of the localization, all localizations being unitary transformations of the canonically delocalized orbitals. This supports the comments of Barone et al.' but illustrates the ineffectiveness of reparameterizing the H atom. It is also sad that CND0/2 is still used2'13 when CNDO/BW is much more effective. 14,15 Other systems can exhibit non-uniqueness in the LMO, such as trichlorovinyl and isocyanate conformers,16 where it is difficult to select the arbitrary criteria to give specific LMO.Such non-uniqueness is unacceptable; bonds should be degener- ate if and only if they are exactly the same, and lone pairs should be degenerate if and only if they are exactly equivalent. If certain criteria can be selected to give equivalent lone-pair orbitals and other criteria to give inequivalent lone-pair orbitals, then there is no mathematical, physical or chemical justification for the resulting LM0.J. One case alone challenges the theory: LMO must be used only within well defined limitations or not at all. -t The authors are grateful to a referee for raising these points.V. Barone, J. Douady, Y. Ellinger and R. Subra, J. Chem. SOC.,Faraday Trans. 2, 1979,75, 1597. P. R. Surjan, M. Rkvesz and Istvitn Mayer, J. Chem. SOC.,Faraday Trans. 2, 1981, 77, 1129. C. Edmiston and K. Rudenberg, Rev.Mod. Phys., 1963, 35,457. 40 I ,O CALI Z A TI 0N 0F M 0L2EC U L A R 0R B IT A L S R. J. Royd and M. A. Whitehead, J. Chem. Soc., Dnlton Trans.. 1972, 73, 78, 81; J. Chem. Soc. A, 1971,3579. W. J. Hunt, P. J. Hay and W. A. Goddard, J. Chem. Phys., 1972, 57, 738. C. Trindle and 0.Sinanoglu, J. Chem. Phys., 1968, 49, 65.'J. H. Liang, Ph.D. Thesis (Ohio State University, 1970).* M. B. Hall, J. Am. Chem. SOC.,1978,100, 6333. B. Levy, Ph. Millie, J. M. Lehn and B.Munsch, Theor. Chim. Acta, 1970, 18, 143. 1 (! W. England, L.. S. Salmon and K. Rudenberg, Top. Ciirr. Chem., 1971, 23,31. I' J. Langlet and J. P. Malrieu, Localizarion and Delocalization in Quantum Chemistry (D. Reidel, Dordrecht, 1976),vol. 11, p. 15. l2 J. D. Goddard and I. G. Czizmadia, Int. J. Quantum Chem., 1977, XII, 133. 13 H. P. Figeys, D. Berckmans and P. Geerlings, J. Chem. SOC.,Faraday Trans. 2, 1981,77, 2091; references therein. 14 M. S. Gopinathan and M. A. Whitehead, Can. J. Chem., 1975, 53, 1343; S. Kishner, M. A. Whitehead and M. S. Gopinathan, J. Am. Chem. SOC.,1978, 100, 1365; A. E. Foti, V. H. Smith, S. Kishner, M. S. Gopinathan and M. A. Whitehead, Mol. Phys., 1978, 35, 111. Is T. J. Tseng and M. A. Whitehead, J. Comput. Chem., 1982, in press. 16 M. A. Whitehead, J. Chem. Phys., 1978, 69, 497. (PAPER 2/584)

ISSN:0300-9238    

DOI:10.1039/F29837900037

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2,1983,79, 41-55 Phase Equilibria in Binary Mixtures Part 1.-Miscibility Gap with Upper or Lower Critical Point BY FRANTISEKVNUK School of Metallurgy, The South Australian Institute of Technology, P.O. Box 1, Ingle Farm, South Australia 5098, Australia Received 14th April, 1982 The general applicability to binary systems of a recently presented, and here extended, new model of liquid-liquid phase equilibria is demonstrated for n-heptane +acetic anhydride, methanol + cyclohexane, mercury +gallium, sodium +ammonia and polystyrene +cyclohexane. New relationships are derived which accurately describe the two-phase equilibria in binary mixtures not only in the critical region but over the whole range of coexistence. By the use of new composition coordinates and order parameters (which are introduced and discussed) the model reveals some novel interrelationships between the coexisting phases.It is shown that the composition and properties of the coexisting phases are mutually linked by the inherent symmetry features of these systems. 1. INTRODUCTION It has been established experimentally that the physical and thermodynamic properties of coexisting phases in a binary system, made up of components A and B, obey the following relations:' as the critical temperature, T,, is approached. In these expressions X1and X2 are the compositions of the coexisting phases (in terms of the component A), a and p are the so-called critical exponents, E: is the reduced temperature I( T,-T)/T,I,Klo and K20 are proportionality constants and X, is the critical composition to which both X1and X2 converge as T -+T,.Relations (1) and (2) hold only at temperatures within the so-called critical range, i.e. a few degrees below T,. For temperatures further away from the critical point it was postulated that additional terms should be added to eqn (1)and (2), which now take the forms and In a new model of phase behaviour in binary mixtures proposed recently3 it was shown that the compositions of the coexisting phases can be conveniently described not only in the critical region but over the whole range of coexistence 41 PHASE EQUILIBRIA IN BINARY MIXTURES by the relations where (7) and The constants Q and M in eqn (7)and (8)are related to Kloand K20in eqn (1)and (2), while n =p and p = 1-a.They can all be obtained directly from the experimental data, as shown in section 5. However, to achieve this extended range of validity, one must express the composition coordinates in a more appropriate form. 2. NEW COMPOSITION PARAMETERS In common practice the composition parameters of the coexisting phases, X, are expressed in molar fractions (x), volume fractions (4) or mass fractions (w) defined by and where A and B are the compositions of A and B, respectively, expressed in molar, volume or mass quantities and AB(xf +xH) = (4: +4B)= (w, +w, ) = 1. One can convert mole fractions (x) into volume (4)or mass fractions (w)via the well known relation where for mole fraction and for mass fraction, pA, pB and MA,MBbeing the densities and molecular weights of the components A and B, respectively.As the result of this non-linear correlation expressed in eqn (11) the shape of the binary phase diagram is unevenly altered when replotted from the (T-x) to (T-4)or (T-w)coordinates (see fig. 1). Once it was thought that the coexistence F. VNUK 43 I I I 1-0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 1.0 x, atomic fraction (full line) or w, mass fraction (broken line) of Hg FIG.1.-Temperature-composition phase diagram of mercury and gallium. The shape of the miscibility gap in the liquid state is more symmetrical when the compositions are expressed as atomic fractions than when expressed as mass fractions.The points show the experimental data of Predel,I3 the full line represents the calculated values given by eqn (34) and (35). curve was more symmetric in volume fraction than in mole fraction, but experience has shown that there is no simple rule about this. Since (x 1-xz) # (41 -&) # (w -w2), except at T = T,, the constants xc, Klo, K20,/3 and a in eqn (1) and (2) take different values with different composition parameters. These shortcomings can be removed if one expresses the composition parameters as and where A and B are expressed, as before, in molar, volume and mass quantities, respectively. With this new set of composition parameters the conversion from molar to volume or mass units (and vice versa) is now a simple and straightforward operation, i.e.ai,Wi =KiCi i = 1,2;j = 4,w. (15) The symmetry of phase diagrams then remains unchanged irrespective of the choice of C, @ or W as the composition coordinates. In fact, we have 16) PHASE EQUILIBRIA IN BINARY MIXTURES and so that the adjustable constants M, n, Q and p in eqn (5) and (6) remain the same in all three systems of units, while @o, Wo=KjCo j=@, W. (18) The conventional fractional units X, # and w are related to the new compositional units C, @ and W by a simple relation 3. ORDER PARAMETER The present model assumes that in a state of complete miscibility the interactions (or contacts) between the like and unlike molecules of A and B are energetically favoured to the same extent.Below T, (or above T, in systems with a lower critical temperature) the like interactions, A-A and B-B, are energetically more favoured than the unlike interactions, A-B, with the result that the molecu!es -4 and B tend to be surrounded more by like than by unlike molecules and the once homogenous mixture separates into an A-rich and a B-rich phase. As the tem- perature interval IT,-TI increases, the interactions between similar molecules become more and more likely, leading eventually to complete unmixing. Under such conditions the order parameter P can now be defined as where rzl is the fraction of like interactions, A-A and B-B, nu is the fraction of unlike interactions, A-B, in the mixture. For a miscibility gap with an upper critical point we have: P=O atT>T, O<P<1 atO<T<T, and generally P r= 1for T << T,.As pointed out by Bragg and Williams4 and Borelius' the order parameter specified in this manner has a temperature dependence where F(nl,nu) is the energy required to replace two unlike groupings A-B with two like ones, A-A and B-B. If F(nl,nu)/k is replaced by M(T,-T)"T('-")and substituted in eqn (8) one obtains f(T)= 1-P(T) (22) where 1/2 P(T)=tanhM($-1)" =1-(5)c2 F. VNUK 45 is the newly defined order parameter. Since P(T) has the same numerical value irrespective of the choice of units for the composition coordinate. 4. SYMMETRY Symmetry features are one of the most appealing aspects of the relationships pertaining to coexisting phases in the critical region.Both the classical and the lattice gas version of the Ising model postulate perfect symmetry in properties of the coexisting phases so that, e.g. in gas-liquid systems, there should be, at all temperatures, the following density relationship: Pc-Pg =PI -pc. (24) In practice this is not so and these symmetry features, if present at all, are confined to a very narrow temperature region in the vicinity of the critical point.6 With binary liquids the situation is even less satisfactory7 and the symmetry requirements are absent even in the critical region, i.e. for mole fraction x 1x1-xcl# Ixc-x2I. (25) Several methods have been suggested to convert the asymmetric shape of the coexistence curve into a symmetric one.? It was claimed that improved symmetry can be obtained if the composition variable is expressed in volume fraction instead of mole fraction.’ While this was the case with many binary mixtures, opposite behaviour has been observed in others.” The present model does possess very distinct symmetry features but on a different level.As implicit in eqn (10) and (11)the two branches of the coexistence curve are completely symmetric with respect to the newly defined “rectilinear diameter” dT).This new kind of symmetry becomes apparent when the phase diagram is replotted with its composition coordinate in 1nC as shown in fig. 2. One can further improve this representation and make both branches of the coexistence curve completely symmetric with respect to C, or W = 1.This is achieved by expressing the composition variable as where x1, 41, w1 and x2, 42, w2 are the molar, volume and mass fractions of the coexisting phases at equilibrium. t Malesinska’ succeeded in converting asymmetric coexistence curves into symmetric ones by expressing the compositions in modified units where q2/ql is a slightly temperature dependent “asymmetry function”. PHASE EQUILIBRIA IN BINARY MIXTURES 340 * 330. In C FIG. 2.-Coexistence curve of n-heptane +acetic anhydride. The compositions are represented in new composition parameter C =A/B instead of the conventional mole fraction x = A/(A+B),where the components A and B are expressed in their respective molar quantities. The two branches of the coexistence curve are completely symmetrical with respect to their "rectilinear diameter".The points are the experimental data of Nagarajan et a1." and the curve represents the calculated values as given by eqn (29) and (30). 304 302 30C T/K 2 98 2 96 294 I -8 -6 -4 -2 0 2 4 6 8 In u FIG.3.-(a) Conventional representation of the miscibility gap in the polystyrene +cyclohexane system. The coexistence curve shows a large degree of asymmetry. 4 is the volume fraction of polystyrene. (b)The same diagram displaying perfect symmetry when the compositions are expressed in modified composition parameters u, specified by eqn (26). (Experimental data of Nakata er d"). By this transformation one can convert even the most asymmetric phase diagram into a fully symmetric one as is shown in fi 3(a) and (b),where the miscibility I!?gap in the polystyrene + cyclohexane system is shown in conventional representa- tion, T plotted against 4, and in its new composition units, T plotted against lnu.F. VNUK 47 5. APPLICATION OF THE NEW MODEL (a) n-HEPTANEtACETIC ANHYDRIDE These new concepts are now applied to the binary system, n-heptane + acetic anhydride, which was experimentally investigated by Nagarajan et a/.’’ between 275.7 K and the critical temperature. The separation temperature was determined on 76 samples ranging in composition from 0.04 to 0.88 mole fraction of n-heptane. In their analysis the authors disregarded the experimental data for those samples where the separation temperature was ca.0.85 IS below the critical point, since here the gravity effects would be expected to have caused “significant distortion of experimental measurements3~.1” In the present reanalysis of these experimental data no prior assumption is made as to the values of T,and x,. These quantities are determined by extrapolation from those experimental data which would not have been affected by the gravity effects. The selection of the experimental data for the analysis (i.e. data with negligible gravity effects) is assessed by the model prediction. According to the model g(T)= (c~c~)”~= c0exp Q(T,-TIP (7) i.e. (C1C2)1’2is an exponential function which can be either monotonically increasing or monotonically decreasing over the whole coexistence range.The experimental data of Nagarajan et a1.l’ show that (C1C2)1/2is monotonically increasing between 275.65 and 341.47 K and decreasing between 341.47 and 341.67 K. This change in behaviour is attributed to the gravity effects, and it is assumed that the gravity- affected data are located in the temperature interval of ca. 2(Tc-341.67), i.e. ca. 0.4 K. Consequently the samples having separation temperature between 34 1.27 K and the critical temperature are not included in the present analysis. The remaining 41 sets of data points were then processed as follows: (i) Molar fractions (xl,x2) were converted into the new composition parameters (Cl,C2) according to eqn (19). (ii) The constants M, n and T, were obtained from eqn (8), which for this purpose is written as: The solution consists of a search for a linear plot of the left-hand side of eqn (27) against In [(TJT)-11 with a value of T, giving the minimum sum of squared residuals.Such a plot with T,= 341.648 K is shown in fig. 4(a). On this linearized plot In (2M) and n are the intercept and slope, respectively. (iii) With this value of T, the constants Co, Q and p were then determined from eqn (7), which is rewritten as Here again, a value of p is sought to give an optimal linear plot of the left-hand side of eqn (7a) against (Tc-T)’. Such a plot, with p = 0.842,is shown in fig. 4(6). The constants obtained by this method are: M = 2,4001 n = 0.2847 T,= 341.648 Q = -0.01326 p = 0.842 Co= 0.9463 PHASE EQUILIBRIA IN BINARY MIXTURES 341.648In (7-1) FIG.4.-n-Heptane + acetic anhydride miscibility gap. (Experimental data of Nagarajan et a/.") (a) Linearized plot of eqn (8) and (27) to obtain constants M, n and T,. Slope = 0.2847. (6) Linearized plot of eqn (7) and (28) to obtain numerical values of Q, Coand p. Slope = -0.0133. giving g(T)= 0.9463 exp [ -0.01326(341.648-T)0.842] (34 lT648 0.2847f(T)= 1-tanh 2.4001 A-from which the values C1and C2are calculated c1,2 = g(T)[f(T)l*l. When these calculated values are converted to the conventional composition para- meters x1 and xz and compared with the experimental data of ref. (lo), they show very good agreement over the whole temperature range under investigation as seen from the deviation plot in fig.5. Note that the calculated value of Tc(= 341.648 K) comes very close to the experimental value as reported by Nagarajan et al. (= 341.672 K), while their value of xc (0.4707) is significantly lower than the calculated value (0.4862). The phase diagram, redrawn within the new coordinates (T against In C) to underline the significance of g(T)and f(T),is shown in fig. 2. (6) METHANOL+CYCLOHEXANE The data on n-heptane + acetic anhydride," analysed in the preceding section, are extensive but do not have the same degree of experimental precision as the F. VNUK 49 I I I I I Id* 11 2 90 300 310 320 330 34 0 T/K FIG.5.-Deviation plot for the compositions of the coexisting phases in the n-heptane +acetic anhydride system [ref. (lo)].Full circles (@) represent the experimentally determined and interpolated composi- tions of the acetic-anhydride-rich phase, empty circles (0)those of the n-heptane-rich phase, in terms of the mole fraction of n-heptane. They are compared with the calculated values obtained from eqn (29) and (30) and converted to mole fractions according to eqn (19). 1 I 1 I-0.2 L I 1 2 3 4 5 6 (318.565 -T)0-708 -6 -5 -4 -3 318.565In (-f--1) FIG.6.-Liquid-liquid phase equilibria in the methanol +cyclohexane system. (Experimental data of Becker et al.'*) (a) Linearized plot of eqn (8) to obtain M, n and T,. ih) Linearized plot of eqn (7) to obtain Co,Q and p. data of Becker et a1.,12 who remeasured the miscibility gap in the system methanol + cyclohexane within 15 K of its critical point.They used the same point-by-point determination method as Nagarajan et al. but employed a more accurate experi- mental technique (turbidity measurements with electro-optical indication). Their experimental data (excluding those within 0.5 K of the critical point) were processsed as outlined in the preceding section. The linearized plots according to eqn (7) and (8) are given in fig. 6(a) and (b),and they show only a very small scatter. The calculated critical constants (Tc= 318.565 K and xc = 0.516) come very 50 PHASE EQUILIBRIA IN BINARY MIXTURES 0.4 0.2 - em T,= 318.565K Y 0- 0 0 0 0- 0 0-0-0 -o-o-oe-.-. 0 . 0-q 0 -0.2 b 0 0 .L ' I I I 1 close to the quoted experimental values, i.e. T,= 318.50f0.05 K and xc= 0.513 *0.002. Complete formulae for the compositions of the two branches are c1,2 = g(mf(n1" where g(T) = 1.068 exp [ -2.924 x lW'(318.565 -T)0.708] and (3 18; 65 1)0.2976f(T)= 1-tanh 2.534 ~-(33) The experimental accuracies claimed by the authors are: kO.05 K in tem- perature and f0.001 in mole fraction. Within these limits there is perfect agreement between the experimental data and those calculated by eqn (32) and (33), as can be seen from the deviation plot (fig. 7). (c) ME R CU R Y + G A L L I U M This is an example of a miscibility gap in a metallic system. The cornpositions of the coexisting liquid phases were investigated by Predel13 and their res ective resistivities near the critical point were measured by Schurmann and Parks!4 The diagram is shown in fig.I. It can be seen that the range of coexistence extends over 170K and at the lowest temperature of liquid coexistence the phases are almost completely unmixed. When the compositions are expressed as mole frac- tions, the diagram is nearly symmetric. This feature is lost when the compositions are expressed in mass fractions. The compositions were read from the diagram and processed as outlined earlier. Again, we have where g(T)= I.072expi-1.114~ 10-3(475.644-T)1~083] (34) F. VNUK 51 0 I FIG. 8.-Deviation plot for the compositions of coexisting phases in the binary system Hg+Ga. (Experimental data of Predel.13) Calculated values obtained from eqn (34)and (35).Ax = 100 (xeXpt-Xcalc). and f(T)=l-tanh2.475(--1)475.644 0.3458. .T (35) The comparison between the experimental and calculated values is displayed on the deviation plot in fig. 8. (d) SODIUMtAMMONIA This binary mixture of two very dissimilar substances has attracted considerable interest, since it represents one of the better known cases of the metal-to-non-metal transition.” Furthermore, it has been claimed that in this system one can observe a clear and well defined transition from the Ising-type critical behaviour near T, to the mean-field behaviour at temperatures ca. 2 K below TC.l6,l7The transition is detected by an abrupt change in (n in the present model) from its Ising value (p= 0.325) to the mean-field value (p= 0.50) at E ~0.01.The experimental data for this ~ysteml~’~~were reanalysed by Das and Greer.l9 They fitted the data with a two-term Wegner expansion as modified for liquid mixtures by Ley-Koo and Green2 [eqn (3)], and claimed “an excellent fit”.19 They achieved this by assigning fixed values to @(0.325),A(0.50) and T, (232.55 K) and determining Kloand Kll by a standard fitting procedure. When the experimental data of ref. (20) supplemented by the interpolated values of Das and Greer” are treated by the present model, a very good agreement is obtained between the experimental/interpolated and calculated data over the whole range of coexistence. The calculated critical parameters (T,= 231.52 K and xc= 0.0404) are virtually identical with the experimental ones (T,= 231.55 K x, = 0.04111, while n =0.389 (as compared to two values of Das and Greer, i.e. 0.5 and 0.325).The compositions of the coexisting phases are given by C1,J= -x1,2 - g(T)[f(T)]’l1-x1,2 (19931) where g(T)=4.232 x exp [-3.313 x 1OP3(231.52-T)1.128] (34) PHASE EQUILIBRIA IN BINARY MIXTURES 0.02 2 % 0.04 -- --0 -y+.ok, c .-Y wE 0.06- 0.94 - e, 2< 0.08- // ;O 231 23: T,= 231.52 K 0.101 0.2 c I 210 215 220 225 230 TI K FIG. 9.-(a) Coexistence curve in Na+NH3 system. Experimental data of Kraus and Lucasse" and interpolated values of Das and Greer'' shown as circles (0).Experimental data of Chieux and Sienko16 are shown as crosses (+) but these were not used in the evaluation of the constants in eqn (36) and (37).(b)Deviation plot showing the extent of scatter between the experimental and calculated values. Ax = 100 (Xexpt -Xcalc). and 0.389 f(T)= 1-tanh 1.685(23;52 1) , (37) The phase diagram, together with the deviation plot, is shown in fig. 9. (e) P OLY STY R EN E +cY CLO HEX ANE" The respective molecular weights of the components in this system differ by more than 4 orders of magnitude and consequently the coexistence curve is one of the most asymmetric, having the critical composition ca. 0.032 volume fraction of polystyrene. Lowering the temperature by 3 K results in a four-fold increase in the concentration of polystyrene in the polystyrene-rich phase, but an eighteen-fold decrease in the cyclohexane-rich phase.At still lower temperatures the solubility of polystyrene in cyclohexane becomes so small as to be of the same order as the width of the experimental error. As a result, the solubilities of polystyrene in cyclohexane can be obtained only with a low relative accuracy [see fig. 3(a)]. However, the new model makes it possible to determine the composition of the cyclohexane-rich liquids with greatly improved accuracy on the basis of the symmetry properties expressed in eqn (26). F. VNUK 53 0 0 -0.10 294 296 298 300 II . "302.0 302.5 303.0 303.5 T/K FIG. lO.-Deviation plot for the compositions (in terms of mole fraction of polystyrene) of coexisting phases in polystyrene+cyclohexane system [ref.(ll)]. Note the change in temperature scale at T> 301 K. Full circles (0)represent the cyclohexane-rich liquid. Calculated values obtained from eqn (38) and (39). A4 = 100 (4expr-Llc). First the function f(T)is derived from those values of 41 and q52 where the experimental results are most reliable, i.e. within the temperature range 0.029 < (T,-T)<2.983 K. This gives (303;53 1)0.3313f(T)= 1-tanh 6.752 ______-The function f(T)is then extrapolated to lower temperatures and the values of 41are calculated from By this method the calculated values are: 0.000 89 at 299.174 K, 0.000 41 at 296.65 K and 0.000 16 at 292.61 K, as compared with the corresponding values of Nakata et a/.," viz.0.0012, 0.0009 and 0.0009. With these values one can now calculate g(T)and obtain a full expression for the composition of both branches of the coexistence curve: where g(T)=3.3993 X exp [-0.4288(303.653 -T)0.516] (39) and (303T653 1)0.3313f(T)= 1-tanh 6.752 A-The comparison between the calucated and experimental values is shown in fig. 10. 6. SUMMARY AND CONCLUSIONS (i) A new model for phase equilibria in binary mixtures has been proposed. In its versatility and applicability to a diversity of binary mixtures and over an extended temperature range it is superior to existing models. It achieves these advantages PHASE EQUILIBRIA IN BINARY MIXTURES with an economy of adjustable constants which deserves repeated emphasis in this summary.Comparing the performance of the present model with that of Wegner’s ex ansion2 as applied to the n-heptane+acetic anhydride system by Nagarajan et al!’ one observes: (a) The difference in “order parameter” (x2-xl) for the coexisting phases in the n-heptane +acetic anhydride system was satisfactorily fitted to eqn (3) by Nagarajan et al. with four constants (ISlo,Kll, p and A) when compositions are expressed in mole fractions and with five constants (Klo,KI1, K12,p and A) when compositions are expressed in volume fractions. In the new model the “order parameter” expressed as ratio (C1/C2)1’2(where C1< C2)can be satisfactorily fitted to eqn (8) with two constants M and n (= p) which have the same value irrespective of whether the composition coordinates are expressed in mole, volume or mass units.(b)The “rectilinear diameter”, $(x1+x2), for the coexisting phases in the n-heptane +acetic anhydride system was satisfactorily fitted to eqn (4) by Nagarajan et al.” with five constants (xc, K20,K21,a, A). In the new model the “geometrical rectilinear diameter”, (C1C2I1”,can be satisfactorily fitted to eqn (7)with three constants Q, p and Co,where Cois the critical composition in the appropriate composition parameters. (ii) The expressions for the “order parameter” [eqn (S)] and the “rectilinear diameter” [eqn (7)]can be readily used for formulating simple relationships between the compositions of the coexisting phases and T, i.e. C2,@2, ui= g(T)/f(T) (41) where C1,a1,W1and C2,Q2, Wzare the new composition parameters related to the fractional composition parameters x1, 41,w~1and x2, 42, w2 by (iii) The above formulations are not only simple but they also reveal a new and intrinsically appealing type of symmetry, so much sought in all analyses of critical phenomena.One can see that the two branches of the coexistence curve are perfectly symmetrical with respect to the new “rectilinear diameter”, g (T)= (C1C2)li2.This symmetry can be displayed in a striking manner if the temperature- composition diagram is plotted with the abscissa in In C, as shown in fig.2. (iv) The new mathematical formulations hold over the whole range of coexistence, including T = T,. This unique feature endows the model with an extrapolative power which enables one to determine the critical. parameters T, and xc by extrapolation of the experimental data obtained at temperatures further away from T, where the disturbances due to gravity effects are less severe or negligible.(v) The model suggests a new and more appropriate expression for the order parameter which is one of the most important but rather elusive quantities in the study of coexisting phases. By relating the order parameter P to the probability of like and unlike contacts in the mixture an expression for the temperature dependence of P is obtained: P = tanhivl($-I)‘*. This expression defines P uniquely, irrespective of the choice of composition coordinate. F. VNUK 55 (vi) Exponent n in eqn (23)is identical with the critical exponent p in eqn (1).This p is considered in current theories to be a universal constant with a fixed value for all binary systems (p = 0.312 from the lattice-model expansion technique and p = 0.325 from the normalization group calculations*’). The present model suggests that p is not a universal but an adjustable constant, and (in the same sense as Tc)has a characteristic value which varies from system to system. It appears, however, that in a given system n (=p)remains unchanged even when the coexisting phases are specified by their respective physical properties (e.g. refractive indices, densities, electrical resistivities, etc.) instead of compositions. -i-(vii) The principles applied to the coexisting phases in miscibility gaps are applicable also to other types of phase equilibria, as described in the following paper, and to other cooperative critical phenomena, as we hope to discuss in subsequent papers.+‘E.g. in the system mercury i-gallium, analysed in section S(c 1, the order parameter ,,75;44 1,)o.345aP = tanh 2.475 ~-when calculated from the compositions of the coexisting phases [ref. (13)], and 0.34’ 6;4 7 34P = tanh 0.630 ____ -when calculated from the electrical resistivities of the coexisting phases [ref. (1411 in the temperature range 469.31 < T/K<476.53. It is seen that in both cases the critical exponent n has the same value within the limits of experimental error. R. L. Scott, in Chemical Thermodynamics, ed. M. L.McGlashan (The Chemical Society, London, 1978), vol. 11, p. 238. F. J. Wegner, Phys. Rec. B, 1972, 5, 4529. M. Ley-Coo and M. S. Green, Phys. Rev. A, 1977, 16, 2483. F. Vnuk, J. Chem. SOC.,Faradny Trans. 2, 1981,77, 1045. W. L. Bragg and E. J. Williams, Proc. R. SOC.London, Ser. A, 1534, 145, 699. G. Borelius, Ann. Phys., 1934, 20, 57,‘J. M. Levelt-Sengers, J. Straub and M. Vincentini-Missoni, J. Chem. Phys., 1971, 54, 5034. A. Stein and G. F. Allen, J. Phys. Chem. Ref. Data, 1973, 2, 443. B. Malesinska, Bull. Acad. Polon. Sci,, Ser. Sci. Chim., 1960, 8, 53 and 61. J. H. Hildebrand and D. R. F. Cochran, J. Am. Chem. SOC.. 1949,71, 22. 10 N. Nagarajan, A. Kumar, E. S. R. Gopal and S. C. Greer, J. Phys. Chem., 1980, 84, 2883. 11 M. Nakata, T.Dobashi, N. Kuwahara, M. Kaneko and B. Chu, Phys. Rev. A, 1978,18, 2683. 12 F. Becker, M. Kiefer, P. Rhensius, A. Spoerner and A. Steiger, 2.Phys. Chem. (Neue Folge), 1978,112, 139. 13 B. Predel, 2.Phys. Chem. (Neue Folge), 1960, 24, 206. 14 H. K. Schurmann and R. D. Parks, Phys. Rev. Leu., 1971,26, 267. 15 J. C. Thompson, Electrons in Liquid Ammonia, (Oxford University Press, Oxford, 1976).16 P. Chieux and M. J. Sienko, J. Chem. Phys., 1970, 53, 566. 17 D. B. Fenner, M. P. Kuhls and D. E. Bowen, Phys. Rev. A, 1978,18, 2707. 18 C. A. Kraus and W. W. Lucasse, J. Am. Chem. SOC.,1922, 44, 1949. 19 B. K. Das and S. C. Greer, J. Chem. Phys., 1981, 74, 3630. 20 S. C. Green, Acc. Chem. Res., 1978, 11, 427. (PAPER 2/629)

ISSN:0300-9238    

DOI:10.1039/F29837900041

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1983,79, 57-64 Phase Equilibria in Binary Mixtures Part 2.-Miscibility Gap with Two Critical Temperatures (Closed-loop Phase Diagram) BY FRANTISEKVNUK School of Metallurgy, The South Australia Institute of Technology, P.O. Box 1, Ingle Farm, South Australia 5098, Australia Received 14th April, 1982 The model outlined in the preceding paper is applied to binary mixtures which feature a miscibility gap with two critical temperatures. By simple modification of the mathematical expressions for the order parameter and the rectilinear diameter, the model accurately describes equilibria between the coexisting phases over the whole range of coexistence. The model has been applied to the following binary systems: tetrahydrofuran +heavy water, 2,6-lutidine +water, s-benzylethylamine +glycerol and bismuth +bismuth trichloride.1. INTRODUCTION In the preceding paper' it was shown that the phase equilibria between the coexisting phases in a binary system A-B can be satisfactorily described over the whole range of coexistence by the following relations: where C1 and C1 are the suitably chosen composition parameters of the A-rich and B-rich phase, respectively, and the functions g(T) and f(T)are expressed by f(T)=l-tanhM The formulations are applicable to binary systems having either an upper or lower critical point. However, it has been known since 19032 that there is a limited number of binary systems which possess not one but two (i.e. upper and lower) critical points. In such systems the two-phase coexistence region is completely surrounded by a single-phase region.Such a closed-loop coexistence curve represents a rather uncommon feature in certain liquid binary systems in which water is often one of the components. A theoretical contribution towards the understanding of this behaviour was advanced by Hirschfelder et ~l.,~on the basis of which simple models were set up by Barker and Fock4 (lattice solution model) and Wheeler and coworkers' (decorated lattice model). A lattice model (based on the Migdal-Kaganoff approximation) was proposed by Walker and Vause.6 A thermodynamic approach to closed-loop phase diagrams, which involves alternative composition parameters, was suggested by Ke hiaian .7 It is proposed to show that the model outlined in the preceding paper can be also readily applied to these systems, i.e.to the closed-loop phase diagram. 57 PHASE EQUILIBRIA IN BINARY MIXTURES 2. “ORDER PARAMETER” AND “RECTILINEAR DIAMETER” FOR CLOSED-LOOP SYSTEMS In mixtures showing a miscibility gap with one critical point the unmixing starts at T, and steadily increases as one moves further from T,. As noted in the previous paper, the order parameter, P, reflects the extent of unmixing. P = 0 for complete miscibility, 0 <P < 1for partly unmixed phases, and P = 1for complete unmixing. In these systems P increases monotonically from zero at T = T, to its maximum value at the other temperature end of the range of coexistence.The situation is different in closed-loop systems. Here the two-phase region is confined within the temperature range bounded by the upper (Tu)and lower (TL) critical temperatures. Within this range the coexisting phases (when cooled from Tu to T,) undergo initial unmixing eventually followed by the reversal of the unmixing process leading again to complete miscibility at TL.The order parameter P, defined as P(T)= tanh ME (5) where E is the reduced temperature, must therefore converge to zero at each of these two extremes of the two-phase coexistence, i.e. at Tu and TL. Between these two temperatures P has a non-zero value which, however, never reaches its numerical maximum, P = 1. Furthermore, the order parameter P(T)must change smoothly and continuously, since the phase boundary itself does not show any sign of discontinuity.A simple functional form of E which includes both critical temperatures and makes P(T)behave in such a manner is E = (+-1)(1-+) . This function gives P(T)= 0 at T = Tu and TL.It reaches its maximum value at One can consider Tm as the temperature at which the reversal of the unmixing process sets in during cooling. Substituting eqn (7) into eqn (6)gives Since O<E,,,~~<~,O<P(T,)<l. The functional form for f(T)is f(T)= 1-tanh M [(+-1)(1-31n. (9) While function (4) is valid over the whole temperature range Tu-TL,function (3) cannot hold over this whole ran 8e since Cy # Ck and one cannot make g(T) converge to Ck at T = TLand Co at T = Tu, even if (Tc- T) is replaced by [(TU -T)(T-TL)l.It is then obvious that g(T)(i.e.the “rectilinear diameter” in the closed loop) must be fitted with two sets of the adjustable constants Co, Q and p, as used in F. VNUK 59 the preceding paper; one set applicable to the lower temperature range and the other applicable to the upper temperature range. The temperature Tp, which separates the closed loop into these two regions, is obtained from a smoothed plot of the rectilinear diameter (C1C2)1/2against temperature. Tprepresents the tem- perature at which d(C1C2)/dT = 0 (i.e.maximum or minimum) or d2(C,C2)/dT2 = 0 (i.e. the inflexion point of the smoothed curve). Generally Tp# T,. Thus gu(T)= Cy exp[Qu( Tu-T)pu] for Tp< T < Tu (104 gL(T)=Ckexp[QL(T-TJL] for TL<T<Tp.(lob) The compositions of the two branches of the closed-loop system are now completely described by the following expressions: and 3. APPLICATION OF THE MODEL Four examples were selected from a relatively small number of closed-loop binary systems. The experimental data were processed by the methods which were outlined in the previous paper. Regrettably none of the known closed-loop systems has been accurately and thoroughly investigated. Reliable data are scarce, since most of the investigations on these systems were made 50-80 years ago. Accurate data of recent origin are usually confined to a narrow range of temperatures in the vicinity of the lower critical point.' (a)TETRAHYDROFURAN + HEAVY WATER The most extensive experimental data on a closed-loop system are those reported by LejEek et al.' on THF and D20.The critical values of Tu(=416.71 K) and TL(=336.88K) for this system were determined from an enlarged plot of their data. Such plots were also used (for this and the following systems) to interpolate the composition of the relevant conjugate phase. The constants M and n were obtained from a plot of In ln[2(C2/Cl)'/2- 11 against ln[(TU/T- 1)(1-TL/T)], which is shown in fig. 1. Tpwas estimated as 373 f5 K and the experimental data above and below this temperature were processed separately. They yielded: gu(T)= 0.8386 exp [5.122 x 10-5(416.71-T)2.0]; 337 K< T < Tu (12a) gL(T)= 1.148 exp[-3.556x 10-2(T-336.88)0.5]; TL<T<337 K (12b) f(T)= 1-tanh 2.723 416.71 0.2691 (124 The calculated and experimental data, together with the deviation plot, are shown in fig.2. PHASE EQUILIBRIA IN BINARY MIXTURES X YXS” ‘I* 1 1 1 I I I I I 1-2 ‘ X -12 -11 -10 -9 -8 -7 -6 -5 -4 In [(41c71 1)(1--33:88)]~_ FIG.1.-Linearized plot of eqn (9)to determine the constants M and n for the system tetrahydrofuran + heavy water. (Experimental data of LejEek et aL9) Open circles (0)represent data points in the upper part of the loop (372 < T/K<416.7), crosses ( x ) those in the lower part (336.9 < T/K< 372.6). n = 0.269. 0.2 0.3 5!5 0.4 0 C .d t; 0.5 2+ VJ 0.6 G 0.7 (b) +e-72 1-h.,?-??L!L+-&-y -:--.A**-40 *+.~ 9* 0’ ”3v8 -1-TP ?+ 0, 1 I I I I I I I F.VNUK 61 I I 1 I I I I 1 1 1 1 0.2 20.4 5: W :0.8 E 2-(61o+o + 0 +oof 2 O-gT :-n-&o-.,-+ A + &,o~O', -2 -00 TPI1 I I I 1 I 1 1 1 1, (b)2,6-LUTIDINE+WATER For this system the region near the lower critical point has been thoroughly investigated.8 The compositions of the coexisting phases over the whole length of the closed loop (which extends from 307.1 to 503.6 K) were established by Andon and Cox.1o Their data can be satisfactorily expressed by the model equations gu(T)= 1.412 exp [1.284 x 10-3(503.G-T)1.22]; 324 K <T <Tu (134 g,(T)=2.314 expr6.4~ 10-2(T-307.1)".42]; TL<T<324 K (13b) [(50;.6 30;.1)]0.334f(T)=1~tanh3.717 ___-1)(1-------The comparison between the experimental and calculated values, together with the deviation plot, is presented in fig. 3.(c) s -BENZYLETHYLAMINE+GLYCEROL This system, investigated by Parwatiker and McEwen," is one of a few known non-aqueous systems with a closed-loop miscibility gap. There are two unusual features in this system: (i) a wide temperature range of coexistence (ca.230 K) and (ii) an almost flat top and bottom of the coexistence curve. However, even this unusual shape of the coexistence curve can be satisfactorily described by the model PHASE EQUILIBRIA IN BINARY MIXTURES 0 300 350 400 450 500 550 600 -r I I 1 I I I (a) 0.2 - 0.8 e, +--xk=0.335 cE '$0.4W - - 0.6 0 ud G .-+ 20.6 2 E0.8G w m -- 0 T,= +x: = 0 592 553.5 c 0.4 '5 2 2 0.2 E w m 1 i FIG.4.-a Phase diagram of s-benzylethylamine +glycerol. Experimental data of Parwatiker and McEwen' i'shown as circles, the values calculated by eqn (14a, b and c) shown as continuous line. (b) Deviation plot. Filled circles (0)represent experimental data of ref. (1l), crosses (+) represent interpolated values. Ax = 1OO(x,,,, -x,,,~). over the whole range of coexistence by gu(T)= 1.45 exp [-0.106(553.5 -T)0.32];417 K < T <Tu (144 gL(T)= 0.504 exp [7.6x 10-3(T-323.2)'.20]; TL< T <417 K (14b)[(55;.5 1)(1--32;.2)]f(T)= 1-tanh 2.446 Experimental and calculated data, together with the deviation plot, are shown in fig. 4. (d)Bi +BiC13 This is a system12 which displays a pronounced retrograde solubility in the liquid state, Such behaviour can be found also in other systems, such as Bi and BiBr3, 13 methylethylketone +butanol,14 butan-2-01 +water,15 etc.In these instances the lower critical temperature is not reached because the unmixing process is terminated by superimposition of another reaction involving one or both liquids and resulting in the formation of a solid phase. If, however, the value of Tu is known, TLcan be reasonably estimated from eqn (7). For this purpose T, was determined from a graphical plot of In In [2(C2/C1)1'2-11 against T. Such a plot for the Bi+BiC13 system revealed that T, =688 K, which gave TL= 513 K. By processing the experimental data in F.VNUK 63 600 700 800 900 1001) 1100 0.2 I I I ;9” 0.4 0.6 8 w m YE: C.-i =0*512-1+ xc 0.6 0.4 .e 0 2v, Q)i /o -0 Y 0.8 0’ 0.2 /do’ O/O 1.0 2 a -2 -a I 1 b? I I 1 600 700 800 900 1000 1100 T/K FIG. 5.-(a) Liquid miscibility gap in the system Bi+BiC13 treated as a truncated closed-loop system (see text). Experimental data of Yasim ef. a1.’* Bi-rich phase shown as open circles (0),BiCl,-rich phase as filled circles (0).Continuous line represents the values calculated from eqn (15~1,b and c). (b)Deviation plot for the system Bi +BiC13. Ax = lOO(x,,,,-xcalc). the manner outlined above, one obtains grJ(T)= 1.048 exp [3.27 x lop3(1048-T)l.’]; 800 K Tu (154 gI,(T)= 6.551 exp [-1.3 12x T -513)1.16]; TL< T <800 K (15b) The experimental and calculated data are shown in fig.5 together with the deviation plot. 4. SUMMARY AND CONCLUSIONS The model presented in this paper considers the closed-loop coexistence curve as being a special type of miscibility gap curve having two critical temperatures. With respect to this crucial difference the expression for the order parameter P(T) is modified to include both upper and lower critical temperatures. With this modification P(T)becomes a reliable indicator of the qualitative and quantitative tendencies of the coexisting phases either towards mutual solubility or towards unmixing over the whole coexistence range. However, for an exact determination of the compositions of coexisting phases at TL<T <Tu one must evaluate also the “rectilinear diameter”, g(T).Unlike the order parameter, g (T)cannot be applied over the whole range of coexistence with just one set of adjustable constants. 64 PHASE EQUILIBRIA IN BINARY MIXTURES Separate sets of values of Co, Q and p must be determined for the upper and lower regions of the closed-loop diagram. The model has been applied with satisfying results to a variety of closed-loop liquid-liquid equilibria. Its suitability for application to liquid-solid and liquid- vapour equilibria in binary systems is being investigated. I thank those who so generously helped in the preparation of this manuscript: R. F. Jones for his assistance with computer programming, D. E. Mulcahy and W. S. Boundy for valuable discussions, and V.Richer for patient typing and re-typing. I also thank the referees for their scholarly but critical examination of both texts. ’ F. Vnuk, J. Chem. SOC., Faraday Trans. 2, 1983, 79,41. * C. S. Hudson, 2.Phys. Chem., 1903, 47, 114. J. Hirschfelder, D. Stevenson and H. Eyring, J. Chem. Phys., 1937, 5, 896. J. A. Barker and W. Fock, Discuss. Faraday SOC.,1953,15, 188. G.R. Anderson and J. C. Wheeler, J. Chem. Phys., 19?8,69,3403 and J. C. Wheeler, J. Chem, Phys., 1975, 62, 433. J. S. Walker and C. A. Vause, Phys. Lett. A, 1980, 79, 421. ’ H. Kehiaian, Bull. Acad. Polon. Sci.,Ser. Sci. Chim., 1962, 10,579 and 585. * E.g. the system 2,6-lutidine+water which was investigated near its TLby J. D.Cox and E. F. G. Herington, Trans. Faraday SOC., 1956, 52, 928, by A. W. Loven and 0. K. Rice, Trans. Faraday SOC., 1963, 59, 2723 and more recently by M. A. Handschy, R. C. Mackler and W. J. O’Sullivan, Chem. Phys. Lett., 1980, 76, 172. P. LejEek, J. MatouS, J. P. Novak and J. Pick, J. Chem. Thermodyn., 1975, 7, 927. 10 R. J. L. Andon, and J. D. Cox, J. Chem. SOC., 1952,4601.11 R. R. Parwatiker and B. C. McEwen, J. Chem. Soc., 1924,125, 1484. 12 S. J. Yasim, A. J. Darnell, W. @. Gehman and S. W. Mayer, J. Phys. Chem., 1959,63,230.13 S. J. Yasim, L. D. Ransom, R. W. Sallach and L. E. Topol, J. Phys. Chem., 1962, 66, 28 and H. Hoskins, K. Tamura and H. Endo, Solid State Commun., 1919, 31, 687. 14 V. Rothmund, 2.Phys. Chem., 1898,26,433.15 T. Moriyoshi, S. Kaneshina, K. Aihara and K. Yabumoto, J. Chem. Thermodyn., 1975, 7, 537. (PAPER 2/630)

ISSN:0300-9238    

DOI:10.1039/F29837900057

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1983,79, 65-76 Temperature Dependence of Emission Spectra and Excited-state Lifetimes in Pure and Nickel-doped One-dimensional BaPt( CN)4s4H20 BY STEPHEN CLARK, PETER DAY,* DAVID J. HUDDARTAND CHARLES N. IRONSIDE University of Oxford, Inorganic Chemistry Laboratory, South Parks Road, Oxford OX1 3QR Received 21st April, 1982 Emission decay curves and time-resolved emission spectra have been measured as a function of temperature for 10 to 50K in crystals of the one-dimensional salt BaPt(CN)4-4H20, both undoped and with 1:650 and 1:65 doping by Ni(CN)i-. The decay curves of the lowest-frequency emission (18400 cm-' at 10 K) are least-squares fitted to obtain the variation of the longest-lifetime component with temperature. The ratio of the intensities of the 18400 and 20 100 cm-' bands in the short (50 ns) time-resolved emission spectra varies with temperature and doping level.The results are interpreted with a three-level model, including radiationless passage from the lowest excited state of Pt(CN):- to Ni( CN)i-. The unusual optical properties of the one-dimensional tetracyanoplatinate(I1) salts were first noticed early in the last century' and were investigated by such well known optical physicists as Brewster2 and stoke^.^ The latter observed the intense green emission from yellow BaPt(CNj4.4H20 crystals, noting the contrast between such solids and Pt(CN)i- in solution, where neither absorption nor emission occur in the visible. More recently it was found that the absorption spectra of the tetracyanoplatinate(I1) salts, whose structures consist of face-sharing stacks of square-planar anions, were dominated by an extremely intense broad band whose frequency varied with the plane-to-plane spacing within the We have argued6 that the simplest approach to the excited states of these ionic molecular crystals is to treat them as neutral Frenkel excitons propagating along the stacks.Our hypothesis is supported by the fact that for at least a dozen salts of this kind the frequency of the intense absorption band decreases in proportion to the inverse cube of the Pt-Pt ~eparation,~.~as required if the energy gap arises from interaction between molecular transition electric dipoles (the Davydov effect).Several thousand cm-' to lower frequency than the absorption, the emission [whose frequency also decreases as R (Pt-Pt)-'I8 has components with polarisation both parallel and perpendicular to the stacks,' the latter being the more intense. Both components shift quite strongly to lower frequency as the temperature is lowered," and since a similar shift occurs under hydrostatic pressure at fixed temperat~re'"~ the temperature-induced shift is probably due to contraction of the lattice. In BaPt(CN)4.4H20 the limiting frequencies of the two components at liquid-helium temperatures are 20 100 (E11 c)and 18400 cm-I (Eic).~',~~Decay curves and time-resolved emission spectra" reveal that the two emission peaks have very different lifetimes which, furthermore, have different temperature depen- dences.The higher-frequency band has an extremely short lifetime (<3 ns) while the other, which cannot be described by a single exponential decay, contains at least 65 EM ISSI ON SPECTRA OF BaPt(CN)4.4H20 one component that becomes very long-lived at low temperature. In this paper our aim is to report the temperature dependence of the lifetimes and the relative intensities of the two components in detail, and to test the applicability of a three-level model to this class of excited state. Concentration quenching by impurities provides a powerful probe of the exciton dynamics in crystals, and it has been known for many years that small quantities of Ni(CN);- quench the fluorescence of the tetracyanoplatinates. l3 As expected, the reduction in the emission intensity is a power-law function of the dopant concentration and mol '/o Ni already have a detectable effect on the inten~ity.'~''' These conclusions are confirmed by more detailed measurements on single crystals'6 which indicate that impurity quenching is confined to the lower- frequency E Ic emission, the intensity of the higher-frequency E 11 c band remaining almost unaffected by impurity concentration. No lifetime measurements have been reported for nickel-doped tetracyanoplatinates, so we also give data for BaPt(CN)4.4H20 crystals containing high (1 :65) and low (1 :650) mole ratios of Ni( CN);-.EXPERIMENTAL Crystals of BaPt(CN),.4H20 and BaPt(Pt,-,Nix) (CN),.4Hz0 were grown from aqueous solution by slow cooling.Individual nickel-doped crystals used for the emission experiments were subsequently analysed by atomic absorption spectroscopy using a Perkin-Elmer 306 spectrometer with an HGA 72 flameless graphite furnace and BaNi(CN),.4H20 as a calibrant. For the more dilute samples the mole ratio of Ni was 2-3 times that of the solution from which they had been grown. Nominally undoped crystals were also analysed and a small concentration of Ni was detected (1:4000). Crystals were mounted in an Oxford Instruments CF204 continuous-flow helium cryostat equipped with an Oxford Instruments DTC2 temperature controller, and cooled by helium exchange gas. Slow pumping in the initial stage of cooling ensures that the crystals do not lose water.Emission was excited by a Molectron DL 200 dye laser pumped by a Molectron UV 400 pulsed nitrogen laser. In the earlier experiments the emitted light was dispersed by a Hilger D331 double monochromator, detected by an EM1 9816QB fast photomultiplier and displayed OR a Hewlett-Packard 1741A 100 MHz oscilloscope, where it was photo- graphed with a Polaroid camera. Photographs taken over a wide range of time domains were fitted by manual curve-stripping to obtain the decay components. In later experiments the emission was dispersed by a Spex 1404 double-grating monochromator, and the signal from the photomultiplier, representing individual decay pulses following each pulse of excitation, was divided among 2000 channels of 10 ns duration by a Biomation 8100 transient digitizer.The contents of the channels were stored on floppy discs in a Research Machines 3802 microcomputer which also controlled the stepping of the analysing monochromator. After averaging a suitable number of laser pulses, the data were displayed on a Hewlett- Packard 7225A plotter in the form of logarithmic decay curves or time-resolved emission spectra. The microcomputer also provided least-squares fits to the decay curves over specified time intervals. Emission lifetimes did not vary with the excitation frequency, which was therefore chosen simply to maximise the emitted intensity. RESULTS In agreement with earlier work we find that even at liquid-helium temperature the decay time of th E IIc emission at 20 100 cm-' is no longer than the laser pulse (5 ns).From the E Ic level at 18400 cm-', however, the decay cannot be described by a single exponential and contained one contribution which lengthed rapidly below 50 K, reaching >10 ps at 10 K. Some typical plots for an undoped crystal, measured between 10 and 50K, are shown in fig. 1. Least-squares fits to the S. CLARK, P. DAY, D. J. HUDDART AND C. N. IRONSIDE 1, 0.5 1 FIG. 1.-Decay of emission at 18 400 cm-' in BaPt(CN)4.4H20 at temperatures between 10 and 50 K. The straight lines are least-squares fits to the longest decay component. longest component are also illustrated, Although the curves are evidently complex, correlation coefficients are better than 97% in every case.As far as the shorter- lifetime components are concened, there is good agreement between the decay times deconvoluted from our photographic data and those in the literature. Thus at 3000K: r1= 3300 ns 72= 600 ns 73= 200 ns (this work) 3110 - 700 680 200 210 [ref. (17j1 [ref. (12j1. There is also evidence (see fig. 2) that in contrast to the longest component the two shorter ones remain almost independent of temperature. Fig. 2 contains data for a 1:2500 nickel-doped crystal as well as an undoped one, and shows that the number of components, and their lifetimes, are not changed by doping. For the longest-lived component at least, this suggestion is confirmed by the more extensive data in fiig. 3, where the lifetimes of the 18 400 cm-' emission of 1:2500 and 1:650 nickel-doped crystals are plotted together with those of an undoped one.The straight-line logarithmic plot of r against 1/T in fig. 3 requires a single activation energy for the decay equivalent to 27* 1cm-' at least between 10 and 50 K. In contrast to the behaviour of the lifetimes, the relative intensities of the two emission peaks do vary with the concentration of nickel-dopant, as well as with temperature. As shown in fig. 4, the intensity of the 20 100 cm-' peak increases EMISSION SPECTRA OF BaPt(CN)4.4H20 0 10 20 30 LO 50 T/K FIG. 2.-Temperature dependence of three decay components in (filled circles) undoped and (empty circles) 1:2500 nickel-doped BaPt(CN),.4H20. relative to the 18 400 cm-' peak as the temperature decreases.At a given tem- perature the higher-frequency peak also becomes progressively more pronounced compared with the lower as the nickel concentration increases. The spectra shown in fig. 4 are time-resolved and represent an integration of the emission from 0 to 50 ns. The ratios of the integrated band areas of the 18 400 and 20 100 cm-l emissions are plotted as a function of temperature in fig. 5 using more extensive data such as that in fig, 4. Clearly the ratio increases with decreasing temperature in almost the same way for the pure and doped crystals, but tends to a limiting value at the lowest temperatures. Aswith the lifetime data in fig. 3, so the logarithm of the band area ratio also falls on a straight line plotted against 1/T, but defines a different activation energy, equivalent to 39f3 cm-' between 20 and 50 K.The significance of these values is discussed below. THEORY AND DISCUSSION We shall examine a simple three-level phenomenological modells for the emission of BaPt(CN)4.4H20 and Ba(Ptl-,Ni,) (CN)4.4H20 which, although it is probably S. CLARK, P. DAY, D. J. HUDDART AND C. N. IRONSIDE 10-4 i x u c)‘c 10-6 0.04 0 08 0 12 KIT FIG. 3.-Temperature dependence of the longest decay component in (filled circles) undoped, (empty circles) 1:650 nickel-doped and (crosses) 1:2500 nickel-doped BaPt(CN)4.4H20. The sizes of the data points represent standard deviations of the least-squares fits. over-simplified, gives a reasonable overall view of the data and acts as a starting point for more elaborate models.In the level scheme in fig. 6 the probabilities ~31 and pZ1per unit time from levels 3 and 2 to level 1 are for radiative processes leading to inverse decay times 75: and r;:, while 3 and 2 are connected to each other by a pathway with a radiationless probability ~32.In the pure salt we assume that these are the only processes but in the doped ones the excitation may transfer to Ni with a probability P2Ni, after which it reaches the ground state by a radiationless pathway. Starting from populations NG and N; the time evolution of the popula- tions is given by dN3ldt = -(p32 +~31)N3+p23N2 (1)dN2ldt =P32N3-(p23 +P2l)N2* Solution of eq (1)depends on further physical assumptions.For example, if there is a Boltzmann distribution of populations between 3 and 2 at t = 0 N$I”Z =~23/~32= (-@32/kT). (2) Then N3(t)=NG exp (-pt) =N; @23/p32) exp (-pt) N2(t)=N; exp (-pt) (3) EMISSION SPECTRA OF BaPt(CN),-4H20 \ 49.8 K I I I I 20 19 18 17 wavenumber/103 cm-' FIG. 4.-Temperature dependence of the fast decay (<5Ons) emission spectra in (a) undoped, (6) 1:6.50 nickel-doped and (c) 1:6.5 iiickel-doped BaPt(CN)4-4H20. This case cannot apply to BaPt(CN),.4H20 since the higher frequency emission has a much shorter lifetime than the lower. On the other hand, if there is no non-radiative back transfer from 2 to 3 (i.e. ~230) then eqn (1)gives= Under the condition that p21<< (p32 +p31), eqn (4) have features in common with what we observe, namely that the decay from 3 is faster than from 2 and that the latter has both fast and slow components.We know from experiment (fig. 3) that S. CLARK, P. DAY, D. J. HUDDART AND C. N. IRONSIDE FIG.4 (cont.) p21is thermally activated, so we should also write 72=p21= p51 exp (-~21/~ (5) with pzl = 2 x lo6s-l and = 27 cm-' from the fit to fig. 3. The other experimental observable available from our experiments is the tem- perature variation of the intensity ratio between the two peaks in the 'time- integrated' emission spectra. The band areas in fig. 4 are related to From eqn (4) it can be shown that if, at a short time following the excitation pulse, the slow component does not contribute to the emission intensity, then the ratio of the intensity of the two peaks is N3(tl)p31 -p31(p32+p31 -p21) (7)N2(tl)(p31+p32) -p32(p31 +p32) EMISSION SPECTRA OF BaPt(CN)4.4H20 20 19 I8 17 wavenumber/103 cm-' FIG.4 (con?.) where we have taken account of the possibility that the population of level 3 can pass to 2 as well as 1 while that in level 2 only goes into 1. We know from experiment that p21 has a much lower probability than ~31 so to the or ~32 approximation that it can be neglected Fig. 5 indicates that at higher temperatures (20-50 K) the intensity ratio between the 18 600 and 20 100 cm-I emissions can be described by a single exponential function, i.e. Although the value of (E32-E31)from fig.5 is not affected by nickel-doping, the limiting ratio R (00) = pz1/p;2 clearly is. In the undoped crystal R (00) = 0.003, while for 1:650 mole fraction Ni, R has risen to 0.006, and for 1: 65 to ca. 0.06. In all three crystals (E32-E31)= 39*3 cm-l. Below 20 K the ratio of the peak intensities can no longer be approximated by eqn (9) and we must use a fuller S. CLARK, P. DAY, D. J. HUDDART AND C. N. IRONSIDE 0.001 f I 1 1 I 0 004 0.08 012 016 K/T FIG.5.-Temperature dependence of the ratio of intensity between the 18400 and 20 100 cm-' emission bands in (filled circles) undoped, (empty circles) 1: 650 nickel-doped and (squares) 1:65 nickel-doped BaPt(CN)4.4H20. The sizes of the data points represent standard deviations.I3-2-P31 P21 -1-Pt Ni FIG.6.-Schematic level scheme for nickel-doped BaPt(CN)4.4H20. Curly lines represent radiationless processes. EMISSION SPECTRA OF BaPt(CN)4.4H20 expression based on eqn (7): If we assume the values of E21,E32-E31 and p031/pi2just given above from the limiting equations are valid at higher temperatures we can then use eqn (8) to find values of all the En, and p;, which reproduce the observed ratio R at the lowest temperatures. For the undoped crystal we find The significance of these numbers is considered later, but first we examine the effect of nickel-doping. Quenching of the emission by transfer to Ni(CN)ip, fol- lowed by radiationless passage to the ground state, could occur in principle from either state 2 or 3.If transfer from state 3 to Ni competes with 3-1 and 3+2 we have which, if p32 is much faster than any of the other processes (as in the undoped crystal), leads to a ratio R * =p31/p32, just as in the undoped crystal. However, the limiting ratio R(m)in the doped crystals is not the same as for the undoped. Consequently we suppose instead that transfer to Ni takes place from state 2 rather than3. Then which, if p32 and p2Ni are fast processes as before, leads to the approximation R @31P2Ni)/@21P32). (12) From experiment we know that R(m)= @gNip;1)/@zlP;2) is ca. 0.006 in the 1:650 doped crystal and ca. 0.05 in the 1:65 crystal. Thus if pi1 etc. remain unaffected by the doping, and retain the values found in the undoped crystal, in the 1:650 crystal we have p;Ni/pil -2 while in the 1:65 crystal this ratio is ca.20. The ten-fold increase in PiNi is quite compatible with the ten-fold increase in the Ni concentration. Our data are also in qualitative agreement with recent measurements of the ratio between the 20 100 and 18 400 cm-' peaks in a much more concentrated crystal of Ba(Ptl-xNix)(CN)4.4H20.19Kasi Viswanath and Patterson" do not report any chemical analysis of their crystal but claim it has a nominal composition x = 0.5. They report the ratio of intensity between the 20 100 and 18 400 cm-' peaks in an integrated emission spectrum as 2 at 250 K and 10 at 125 K, an increase which is certainly consistent with a fifty-fold increase in Ni concentration over our most concentrated sample. No detailed measurement of the temperature depen- dence of the intensity ratio was reported, but is increases with decreasing tem- perature as we find. S.CLARK, P. DAY, D. J. HUDDART AND C. N. IRONSIDE 75 -'l'he three-level model, combined with transfer from the lowest excited state to the Ni impurity, gives a reasonable physical picture of the major processes occurring in the one-dimensional tetracyanoplatinates, although undoubtedly other states play some part. For example with three levels we cannot explain the existence of a process intermediate in time-scale between the fastest and slowest, as shown by fig. 2. However, it does explain the fast decay of the higher-frequency state and the composite fast and slow components of the lower-frequency one, together with the way in which the relative intensities of two time-resolved emission bands vary with temperature and nickel-doping level. It remains to consider the magnitudes of the transition probabilities and activation energies, and their relevance to the assignment of the various excited states.Because the limiting slopes of <lie logarithmic plots of 7 (fig. 3) and R(T)(fig. 5) against 1/T are different, all thrc,h transitions 3 -+ 1, 2 -P 1 and 3 + 2 must be thermally activated, as shown by eqn (13). The values of the activation energies are all low and presumably originate from low-frequency vibrations. The fastest of the three pracesses is the radiationless one, 3+2, and it is interesting to note that this, and not the radiative probability of 3+ 1, is the reason for the rapid depletion of state 3 and the short decay time of the higher-frequency emission.In fact, according to our fit, pgl is actually a little smaller than p;'. In view of the relatively low radiative probabilities of both states 2 and 3 they are most likely to have a triplet parentage, rather than 2 being a triplet and 3 a singlet, as previously proposed.' From the filled ulg(z2), e,(xz, yz) and b2&y) orbitals of Pt(CN)i-, which are mainly localized on the metal atom, the lowest- energy excited configurations are formed by transferring electrons to empty orbitals of u2, symmetry, either of CNT* type or Pt 6p,. The terms which result are 1'3A2u, and 1'3Blu.Spin-orbit coupling is very important in Pt spectroscopy (T~~*73EU = 3500 cm-') and mixes the singlets and triplets so that, for example, there are four E, double group terms2' which will be seen when the electric dipole vector is parallel to the ,molecular plane, as in our experiment.According to the best fit to the magnetic circular dichroism spectrum of Pt(CN);- the lowest in energy of these is the one having predominantly 3A2uparentage.20 Next above it comes a B1,state originating from 3Eu,which is forbidden in both xy-and z-polarization, and then an A2, formed from an admixture of 3Euand 'Az,, which is z-polarized. The frequency range spanned by these three states is cu. 3000cm-', which is quite compatible with the separation between the two emission peaks in the BaPt(CN)4-4H20 crystal. Thus we assign the lowest emission band (state 2, E Ic, 18400 cm-' at 4 K) as EJ3A2,)and the higher one (state 3, E IIc, 20 100 cm-l at 4 K) as A2J3EU, On this interpretation the fast process 3+2 is the result of spin-orbit coupling, which brings about a mixing of 3A2uand 'A2,.Finally, it is worth remarking that the symmetry labels used here are those of the molecular point group D4hand not of the crystal factor group. Since the emission tands are all broad, however, we prefer to consider the spin-orbit perturbation on the molecular states as more important than any factor-group splitting, at least for these nominally spin-forbidden transitions. Further details about the energy trans- fer in this class of one-dimensional crystals must await more refined time-resolved spectral measurements.We are grateful to the S.E.R.C. for financial support. L. Gmelin, Schw. J., 1822, 36, 231. D. Brewster, Rep. Brit. Assoc. Ado. Sci.(20thMeeting), 1850, 5. 76 EMISSION SPECTRA OF BaPt(CN)4-4H20 C. G. Stokes, Philos. Trans. R. SOC. London, 1853, 143, 396. S. Yamada, Bull. Chem. SOC.Jpn, 1951, 24, 125. C. Moncuit and H. Poulet, J. Phys. (Paris), 1962, 23, 353. P. Day, Inorg. Chim. Acta Rev., 1968, 3, 81. P. Day, J. Am, Chem. SOC.,1975,97, 1588. H. Yersin and G. Gliemann, Ann. N. Y.Acad. Sci., 1978, 313, 539. M. Stock and H. Yersin, Chem. Phys. Lett., 1976, 40, 423. 10 P. Day and J. Ferguson, J.Chem. SOC., Faraday Trans. 2, 1981, 77, 1579. 11 Y. Hara, H. Shirotani, S. Ohashi, A. Askumi and M. Minomura, Bull. Chem. SOC. Jpn, 1975, 48,403.12 V. Gerhardt, W, Pfab, J. Resinger and H. Yersin, J. Luminescence, 1979, 18/19, 357. l3 P. Bergsoe, Nord. Kemikenmode, 1939, Farh. 5, 193; Chem. Zentralbl., 1942,II, 2564. 14 IS. Paine, Part I1 Thesis (Oxford University, 1976).15 P. Day, J. Mol. Struct., 1980, 59, 109. l6 W. Holzapfel, H. Yersin and G. Gliemann, J. Chem. Phys., 1981, 74, 2124. l7 Y. Hara, Chem. Lett. (Jpn), 1975, 1063. 18 B. di Bartolo, Optical Interactions in Solids (J. Wiley, New York, 1968), p. 442. l9 A. Kasi Viswanath and H. H. Patterson, Chem. Phys. Lett., 1981, 82, 25. 211 S. B. Piepho, P-N. Schatz and A. J. McCaffery, J. Am. Chem. SOC., 1969, 91, 5994. (PAPER 2/659)

ISSN:0300-9238    

DOI:10.1039/F29837900065

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1983, 79,77-90 The Thermodynamics of a Liquid Lens BY JOHN S. ROWLINSON Physical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ Received 17th May, 1982 The conditions of equilibrium of a small lens of one liquid at the surface of another are obtained in terms of the surface and line tensions. Gravity is neglected. The intersection of the three surfaces of tension, which is also the line of tension, is separated from the surfaces and lines on which surface and linear adsorptions vanish by distances of the order of the range of the intermolecular forces. These separations determine how the surface and line tensions depend on curvature. Experimental and theoretical values of the line tension, although uncertain, allow estimates to be made of the size of systems in which these effects are appreciable.1. INTRODUCTION If a drop of one liquid is placed on the surface of another, heavier, liquid with which it is immiscible, or only partially miscible, then it can either spread over the surface, wetting it completely, or it can remain on the surface as a small lens or bead. Spreading or wetting is the state of equilibrium if the liquid-vapour surface tensions of each of the liquid phases, a and 0, differ by an amount equal to the liquid-liquid surface tension (Antonow's rule'). If surface tension is denoted 0, and if the vapour phase is labelled y, Antonow's rule is that = gPY +g"P (1.1) where all tensions are positive. This rule does not hold for many liquid pairs, e.g.higher n -alkanes on water, for which gay<gBY+gQP; uPy<gay+gaB. (1.2) It is these systems, in which the drop of the lighter liquid (say, phase a) remains as a lens or bead, that are the subject of this thermodynamic analysis. Relations are obtained between the tensions in the three surfaces and in the three-phase line of their intersection. All four tensions depend on curvature in a way that is determined by the three surface adsorptions and the linear adsorption at the three-phase line. Section 2 is a brief recapitulation of the analysis of a system of two phases separated by one spherical surface, as carried out by Gibbs, Tolman, Koenig, Buff, Hill and Kondo.2 This work and the thermodynamic analysis of the tension in straight three-phase lines has been reviewed in a recent book3 whose methods and notation are followed here.Section 3 sets out the mechanical conditions of stability of the three-phase system in the absence of gravity. Section 4 is the thermodynamic analysis, the conclusions of which are summarised in Section 5. 2. A SPHERICAL DROP From a spherical drop of phase a, surrounded by phase p, we choose a system bounded by inner and outer radii R" and RBand conical walls that enclose a solid 77 THERMODYNAMICS OF A 1,IQUID LENS angle o. The boundaries serve only to define the system and do not affect the fluids. The two phases are deemed to be separated by a dividing surface, R"', the choice of which then fixes the values of the volumes, V" and V', and the surface area, AaD.An infinitesimal isothermal change of the Helmholtz free energy of the system can be written, in the usual symbols, dF=-p" dV"-p'dV'+u"'dA"'+C""dR"'+p*dn (2.1) where Cap,the coefficient of the curvature term, is an extensive function, and where p and dn are abbreviations for the sets of chemical potentials p1, .. . ? p, and changes in amounts dnl, . . . ,dn, of each of the c components. Only isothermal changes are considered throughout this paper. Integration of eqn (2. l), by changing the solid angle o,gives F = -p"V" -p'V' +u-"'A"' +p n. (2.2) The condition that F be invariant to the choice of dividing surface leads to the equations 2u"'-[ du'"] -C"' (2.3) ~-R""-dR"' -AffB where square brackets are used to show that the derivative is only the notional change that follows from a displacement of the arbitrary dividing surface in a physically unchan ed system.Since p" -pB is also invariant to such a displacement it follows that u"'is formally a function of R"'. If the surface of tension is defined as the dividing surface, R:'? at which u"' acts, then, by Laplace's equation, where the subscripts s denote quantities evaluated at the surface of tension. Tolman and Hill showed that the formal derivative [du"'/dR"'] could be expressed in terms of the adsorptions I?"' and the real change of u"' with R"' in response to some change in the potentials p; or where r(s)are the absorptions at the surface of tension. In a multicomponent system of two bulk phases, the temperature and (c -1) of the chemical potentials can be chosen independently. If one of the phases is a drop of variable radius then the remaining potential can be chosen to determine its size.The first equation of the set (2.6) is therefore At fixed values of p2,. . . ,p,, and so J. S. ROWLINSON At a plane surface a change in the choice of the height z of the dividing surface leads to a change in the values of the adsorptions; [dI'"'/d~] = -(pa -p'). (2.10) The right-hand sides are constants and so the equations can be integrated. The adsorption of component 1 at the surface of tension, I'Yf3s,, is proportional to the distance of that surface, zs, from the dividing surface, 21, on which l?YP = 0; that is, r%= (6-P%L 81 =21 -2s (2.11) when height is measured from the a to the p phase.Hence, correct to terms of the order of (SlIR,"'), (2.12) and by integration (2.13) where c2' is the tension of the plane surface at the same values of T, pz, .. . ,pc. In a one-component system there is only one potential and eqn (2.13) is then the unique equation that describes how a,"' changes with real changes in RF', the radius of the drop measured at the surface of tension. 3. CONDITIONS OF MECHANICAL STABILITY This section is an examination of the mechanical equilibrium of a system in which three fluid phases, bounded by spherical surfaces, meet in a circular line. In general the shapes of the phases are determined both by the effects of surface and line tensions, and by gravitational forces.The former dominate the physics of small systems, the latter of large ones. The change from one regime to the other occurs for systems whose linear dimensions are comparable with the capillary constant, a, defined by a2=2ca'/gjma *pa-m' -pBI (3.1) where g is the gravitational constant, and ma and mp are the molar masses in the two phases whose molar densities are pa and p'. For water in equilibrium with its vapour at 0 "C and at the surface of the earth, a =3.9 mm. The distinctive effects of curvature on surface and line tensions are found only in systems much smaller than this and so the effects of gravity are ignored in this paper. In its absence, and in the absence of other external fields and boundary conditions, only spherical surfaces are formed; that is, only for a system in which the phases are bounded by parts of spherical surfaces is the free energy a minimum with respect to possible geometrical rearrangements.Consider, therefore, a system in which three fluid phases, a, P and y, meet at three spherical surfaces, ap, ay and Py, which in turn meet at a circular line, aPy. Fig. 1is a section of half such a system taken along a diameter of the three-phase circle, and perpendicular to the tangent at the end of that diameter. The three-phase line appears as the point of intersection of the three curved lines that are the sections of the spherical surfaces. Let the phases be labelled so that pa >pp >py. Since pressure is higher on the inside of a spherical surface it follows that two of the curved lines in the figure must be curved in one direction and the third in the THERMODYNAMICS OF A LIQUID LENS FIG.1.-Half the cross-section of a lens of phase a,separating phases and y. The three heavy lines are the sections of the surfaces, and meet at a point which is the section of the circular three-phase line. The radii of the surfaces are R”’, Ray and R”, and of the circle, R. The angles between the phases are angle a in phase a,etc. other, as shown. Phase a is bounded by two concave surfaces and phase y by two convex. Denote also by a, P and y, the angles at the three-phase line between the pair of surfaces (ap,ay),(ap,Py) and (a?,Py). Let R“’, Ray and ROYbe the radii of the surfaces, and R (=R a’y, but the superscript is omitted for simplicity) be the radius of the three-phase line.If three spheres are to cut in a circle then their centres must be collinear; that is, if the order of the centres on the common line is (a@),(ay),(Py),then R “’R sin a -R“’R OY sin P + R “‘R sin y = 0. (302) An obvious requirement for the stability of the fluid on the three-phase line is that none of the angles a, P or y shall exceed &r. This restriction, together with the convention adopted for the labelling of the phases, rules out any ordering of the four centres, (ap), (ay),(Py) and (aPy) in which (ay)is not between (ap)and (Py),and so restricts us to the four cases: by) (Pr) 2 (4(4 (Pr)1 (4) (Pr) 4 (aPr>(4)(Pd.3 (CUP)(aPr>(4 (4 J.S. ROWLINSON (3) (4) -BY * BY FIG.2.-The four stable configurations of a lens system. An example of each is shown in fig. 2. It is hard to give unambiguous names to the four cases since their shapes change with changes in the four radii. As drawn, case 1shows a large drop of phase a with a lens of phase p. The surface between lens and drop becomes flat if Rapbecomes infinite. Case 2 is two touching drops in which the common surface is again flat if Rapis infinite. If RBTbecomes infinite it represents a bead of phase (Y resting on a plane surface of phase p. Case 4 is a bead of phase a ‘below’ the fly surface. Case 3 is a lens of phase a in or on the Py surface, and isthe example chosen for analysis in the next section.The analyses of other cases would differ only by trivial changes of sign. Consider now the conditions of mechanical equilibrium of an element of fluid on the three-phase line. It is acted on by the three surface tensions, each a force per unit length of line, and by a fourth force that arises from the curvature of the three-phase line. This line is subject to a tension r, which may be positive or negati~e,~and if it is a circle of radius R then an element is subject to a force per unit length of r/R directed towards the centre of the circle: The presence of this force complicates slightly the classification of the previous paragraph but because it is generally weak, (7/R)<< (T, the complication is ignored there.The force must, however, be included in a quantitative analysis. THERMODYNAMICS OF A 121QUID LENS The force per unit length on an element of the three-phase line is therefore “Po a13 +eQyuay+e’”u’” +e””(T/R) = 0 (3.3) where eQ’ is a unit vector lying in the ap-surface where it meets the three-phase line, perpendicular to that line and directed away from it; and where eaPvis a unit vector along the radius of the three-phase circle and directed towards its centre. The ‘vertical’ and ‘horizontal’ components of eqn (3.3) yield, with the aid of eqn (3.2), the two independent equations, R R R (3.4) Rrp)+gay cos (sin -)R cos (sin-’ p)+Rera' cos (sin -1 R Ray --= 0. (3.5) These equations have been derived by macroscopic mechanical arguments but can be interpreted microscopically if the radii are those of the surfaces and line of tension since these are, by definition, the radii ai which the mechanical conditions of equilibrium hold.The first, eqn (3.4), is the2 merely a linear combination of the three Laplace equations, eqn (2.4); that is, it reduces to the identity (p“ -p’)-(p” -p’>+(p’ -p’>=O. (3.6) 4. THERMODYNAMIC ANALYSIS OF THE LENS The system is that shown in section in fig. 1. By rotating this figure about its line of centres through an angle 4 we generate a three-dimensional system. A sphere of radius RY( >R“’ +Roy) centred, say at Py provides an outer boundary to the system and makes Vyfinite. The three volumes V”, V’, Vy,the three areas, A“’, AaY, A”, and the length of the three-phase line, L, are all proportional to 4.The differential form of the free energy, cf. eqn (2.1), is +uffY&7 = -pa dV” -p’ dV’ -py dVY +g“’ u”’ dA”” +(T” ~A’”+TdL+C”’ dR“’+CffY dRaY +C” dR +D dR + p -dn. (4.1) [The system as drawn is not of complicated enough shape to allow all these variables to be changed independently, but this complication could easily be achieved by adding additional boundaries, e.g. a boundary radius Rplying wholly within phase p. Such complication of the geometry would not affect the thermodynamic equations below; it suffices that eqn (4.1) includes all the relevant physical variables as long as all surfaces remain spherical.] By changing this equation can be integrated at fixed c~rvatures to give F=-p“V“ -p’VV”-pYVY+~ffPAnP+~“YA“Y+aaYABY+~L+p~n.(4.2) Three dividing surfaces and one dividing line must be chosen before all volumes, areas and length are fixed. However in a given system the three spherical surfaces must be centred on the three fixed points ap, a:’, and Py of fig. 1. This means that only two of the four radii R (IB, R “”, R ”and R can be chosen independently; the other two are then fixed by the requirement that the surfaces and circle define a common three-phase line. We consider therefore the notional change of free J. S. ROWLINSON energy that follows from an arbitrary displacement of the point on the three-phase line, shown in the section in fig. 1, in the plane of that figure. As before, sqliare brackets show notional changes in a fixed physical system.Let X and Y denote any two of R"", Ray, RBy and R, or any independent pair of linear combinations of them, so that we have [aFlaxl, = [a~/a~]~ (4.3)= 0. From eqn (4.2) (4.4) and similarly for differentiation with respect to Y. The right-hand side of this equation can be evaluated from the geometry of the figure. Thus if X =R Or, Y =R, so that the displacement of the three-phase line is parallel to the line of centres, the right-hand side is From eqn (2.3) it follows that the first three terms on the left-hand side of eqn (4.4) are the same as the first three of eqn (4.5), so that If the three surface radii are those of the surfaces of tension then the right-hand side of eqn (4.6) vanishes, by eqn (3.4).Similarly, if X =R and Y is chosen so that the displacement of the three-phase line is perpendicular to the line of centres, then an equation similar to eqn (4.6) is generated but with a right-hand side derived from eqn (3.5). These results can be combined in a general equation [dT] = [dX]* {effPuffB +eaYosy+e""(r/R)}, (4.7)+effYuaY where dX is any vector displacement in the plane of the section, fig. 1. At the surfaces and line of tension the right-hand side of eqn (4.7) vanishes; that is, [drIs= 0 (4.8) which is the analogue of the second part of eqn (2.4). Thus the placing of the dividing surfaces at the surfaces of tension leads not only to the surface tension but also to the line tension being independent of the arbitrary position of the THERMODYNAMICS OF A LIQUID LENS dividing surfaces, and so a measurable physical property.This is the first useful result of this section. From the differential form of eqn (4.1), aVP[$1 -Pa (5)= -PB(,x)-P"(%) Y =O. (4.9) Subtraction from eqn (4.4)gives (4.10) which, from eqn (2.3),gives (4.11) the analogue of the second equation in eqn (2.3). It follows that C and D all vanish at the surfaces and line of tension C:B= C,""= CFy=D, = 0. (4.12) This second result is a corollary of eqn (4.8). The change of line tension r with a real change of the radius R can be found by subtraction of eqn (4.1)from an expression obtained by general differentiation of eqn (4.2),viz. dRffP+C"' dRffY+ CBYCffB dRPY+D dR = -V" dp" -VPdp@-Vydp' +AaPduff' +AffYdoffY +APYduPY+L dr +n dp.(4.13) This is a generalised Gibbs-Duhem equation. In it we substitute V" dp" = V"p" * dp = n" * dp etc. (4.14) where n" are the molar amounts in phase a, for an arbitrary choice of dividing surface. Similarly, from eqn (2.5), AffBdoffP= CffPdRffB-n"' -dp etc. (4.15) J. S. ROWLINSON 85 FIG. 3.-The line with an arrow is the vector displacement of the dividing surfaces for three planar surfaces meeting in a straight line. The planes and line are shown in section. where n"' are the surface excesses at the cup surface. Hence, from eqn (4.13) D dR =Ldr+n'@' +dp (4.16) where naar are the linear excesses at the three-phase line; n =riff +nP+nY+nff~+nnY+n"+nn"".(4.17) At the line of tension D is zero, eqn (4.12), so dTs=-A(,) * dp (4.18) (4.19) are the linear adsorptions at the line of tension. Eqn (4.18) expresses the change of line tension with changes in the potentials p. To relate this to the change of r with the radius R we need the equivalent of eqn (2.10) for the three-phase system. As before we aim only to calculate the term in r of order R-l, since higher terms have no physical Consider therefore three planar surfaces meeting at a straight line with angles a, p and y in the phases carrying these labels. A notional shift of the dividing line by a vector dX perpendicular to the three-phase line (shown in section in fig. 3) changes the volumes and areas, but not the length of this line.From [dni]=O,where ni, the amount of component i, is given by eqn (4.17), and from n y= Vap:, etc. n y' =AaBry', etc. nq" =LAi, (4.20) we have AaP[dry'] +A""[dI':q +A "[drfq +L[dAj] =-py dV"-pf dVDdp? dVY-~~sdA"~-~~Y~"Y-~fY~BY(4.21) where each change is for a given notional shift [a].Calculation of the changes in volumes and areas and use of eqn (2.10) gives, for each component i, (4.22) which is the analogue of eqn (2.10). This equation can be integrated from a point on the circular line of tension, where Aj = Ai(s),to a point on the circular line on which A, = 0. If Xiis the vector lying in the plane of the section shown in fig. 1, THERMODYNAMICS OF A LIQUID LENS and directed from the point on the line of tension to that on the line of zero linear adsorption, then, for component 1, =-X1-(e"Brr;f,)+e"Yr~,Y,I+eFYT~~,) (4.23) which is the analogue of eqn (2.11).To go further, and obtain a simple analogue to Tolman's eqn (2.13), requires the assumption that the orientations of the four vectors, e"', e"', ear and eaBy, remain fixed on changing pl, and so on changing the four radii. This is a reasonable assumption, since such changes result only from the differences of the changes of a"' etc. with R,"' etc, We know that these changes are small, of the order (S?'/RPB), and if, as is likely, the lengths S1 are all of the same sign and of similar sizes then the changes of the angles may be negligible. It is shown below [see eqn (4.34)] that the condition that the angles are unchanged is (4.24) or (4.25) It follows from eqn (2.9) that eqn (4.24) hdds exactly to the first order of the departure of plfrom its planar limit (P~)~.With this assumption of constant angles we can develop the analogue of eqn (2.13) by scalar multiplication of the force-balance eqn (3.3) by XI,and differentiation with respect to pl.Use of eqn (2.7) then gives From eqn (4.23) (4.27) where El =-(x1* eUoy). (4.28) From eqn (4.27) and the first of eqn (4.18) (4.29) Integration of this equation from a straight line of tension (R, =CO), to an arbitrary radius R,,gives, to O(&I/R,), 5--I--El (4.30) 703 RS since ~1 has a well-defined limit as R, becomes infinite. This equation, the third result of this section, describes how the line tension changes with curvature in response to a change in the system in pl,which, in turn, induces changes in the radii of the surfaces and line of tension.It is the analogue of Tolman's equation (2.13). It cannot usefully be reduced to a one-component form since, except J. S. ROWLINSON for helium at its lambda-point, we cannot have three isotropic fluid phases in equilibrium in a one-component system. In the simplest useful case, a binary system of two immiscible or partially miscible liquids in equilibrium with their vapour, there are no degrees of freedom at a fixed temperature if the phases are separated by planar interfaces; that is, the two potentials have fixed values, say (~1)~and (p&.The equations above, eqn (4.14), (4.15) and (4.18),for component 1, then describe the changes of pa etc., etc. and 7, as is changed from (~1)~.CT:~ These changes are accompanied by the changes in R,"" etc., and R, that are required for mechanical equilibrium to be preserved; that is, for the first part of eqn (2.4)and for eqn (3.3)to be satisfied. If we do not impose the assumption that the changes of angle are negligible we can still determine the change of r with R, but the results are less simple. From the results of Section 2, (4.31) The last term vanishes in the planar limit. An expression for the change of R with p1follows by differentiation of eqn (3.5).Its limit is where (4.32) cos cup =eaS eaBy etc. (4.33) The condition of constancy of angles is that, (4.34) The limiting form of eqn (4.31) and (4.32) satisfy eqn (4.34) if eqn (4.24) or (4.25)hold.In the general case the length E~ of eqn (4.30)can be defined by (4.35) This ratio can be found without approximation from eqn (4.18) and (4.23), which supply the numerator, and eqn (4.32),which is the denominator. It reduces to eqn (4.28) under the condition that eqn (4.26)holds. General expressions for the changes of angles follow from eqn (4.32) and the limiting form of eqn (4.31), ~since sin cw is RJR:~, etc. 5. DISCUSSION The analysis of the last section shows that the tension of a curved line is a measurable physical property if it is defined at the line of tension, which coincides with the intersection of the three surfaces of tension.The line tension depends formally on the radius of this line of tension in a similar way to that in which surface tension depends on the radius of the surface of tension. It is useful to establish a scale of lengths, based on the equations above, which determines which parameters are the relevant ones for any given system. The first THERMODYNAMICS OF A LIQUID LENS scaling length is the capillary constant of eqn (3.1), which is of the order of to 10-2m. For systems larger than this the earth's gravitational field dominates the mechanics, and surface and line tension are unimportant. The second length of interest is the ratio T/U. For soap solutions U= 5 x N m-', and experimental value^^.^ of T range from to N.This ratio is therefore ca. 2 x lo-' m (200 A) or less, according to the size of T. Thus it is only for small lenses and beads that the effects of line tension are appreciable; (T/R,(T)is at most 1/20 for a lens of diameter lop6m or 1pm. Between 1pm and 1mm we can treat capillary problems by surface tension alone, ignoring gravity on the one hand and line tension on the other. There are, however, physical systems of interest for which the characteristic length is less than 1pm, and so for which line tension cannot be ignored. Thus thin soap films (Newton's black films) are m thick, and are, therefore, systems in which line tension is important, and, indeed, in which it has usually been measured. The films that surround the particles in micro-emulsions and lipid bila ers in biological membranes are also structures 2with characteristic lengths of 10- m or less.Nucleation is a phenomenon in which surface properties are important in systems of characteristic size of lop9m. A third scaling length is provided by the distances S of eqn (2.11) and E of eqn (4.30) or (4.35).We have no measurement of these but clearly they must be of the order of the range of the intermolecular forces. A recent exact theoretical calculation4 for a solvable model yielded S = -$d at T = 0, where d is the molecular diameter. Thus these lengths are only ca. 5 x lo-" m. Only for a system in which the radii were of the order of lo-' m could one detect the changes of 0with radius.Recent experiments of Fisher and Israelachvili7 are close to this limit but have not yet established the effect beyond doubt. The effect of curvature on line tension is likely to be even less than that on surface tension, for two reasons. First, the distance E of eqn (4.30) is the scalar product of two vectors, X and the unit vector e"", and so may be numerically considerably smaller than (XI,which is probably comparable with 8. Indeed if the circles of zero linear adsorption and of the line of tension have the same radius, but are displaced from each other along the common line of centres in fig. 1, then X is orthogonal to e"", and so the approximated value of E is zero. The second reason for the smaller effect is that the numerical coefficient has fallen from 2 in eqn (2.13) to 1in eqn (4.30), a consequence of the change of dimensionality from a surface to a line.If a set of three-phase lines were to meet at a point then there would be expected to be a point-contribution to the free energy, although it could now scarcely be called a tension. This contribution comes from an entity of zero-dimensionality for which curvature has no meaning. There would therefore be no analogue of eqn (2.13) or (4.30); there would be a characteristic length, but its coefficient would be zero. Finally it is useful to compare these thermodynamic results with earlier treat- ments of curved surfaces and lines. Buff and Saltsburg' analysed similar systems but in terms of the local stress at the surfaces and three-phase lines.They were not restricted to spherical surfaces. Their equations remain as formal expressions, and it is now apparent3-' that there are difficulties of principle in the interpretation of such equations that arise from the lack of a unique definition of the stress tensor in highly inhomogeneous systems. Boruvka and Neumann' analyse with great generality the thermodynamics of systems of arbitrary (i.e.non-spherical) curvature. Their methods are more general and more abstract than those of this paper, and their useful results correspondingly fewer. Indeed there must be some doubt about the correctness of some of their J. S. ROWLINSON 89 equations since their expression for the energy, the equivalent of eqn (4.2) for the free energy, contains terms of the form C"'R"' etc.and DR. These arise because in attempting to integrate the equivalent of eqn (4.1), they carry out integrations over d(R"')-' etc. and d(R)-'. Energy and free energy are, however, not homogeneous first-order functions of curvature and such integration is impr~per.~-~ Their problems may arise from attempting to treat systems of arbitrary curvatures, not of spherical curvatures, without introducing explicitly an external field, or boundary conditions, which stabilise such curvatures. They have no convenient variable analogous to the angles o of Section 2 and 4 of Section 4. Navascues and Tarazona" discussed a system related to that discussed here, namely a drop of liquid sitting in a cap-shaped depression, which it just fills, in an otherwise planar solid.They considered only the horizontal balance of the forces, i.e. eqn (3.5) but not eqn (3.4), and so did not obtain the general result that the line of tension is the intersection of the three surfaces of tension. They state eqn (4.18) and obtain the change of T with R for their specialised system. Their expression, like eqn (4.35), is not easily interpretable, and the introduction of the assumption that the angles are unchanged with R"' etc. apparently does not lead to a simple equation such as eqn (4.28). They have also developed" a molecular theory of line tension, but only for the straight-line contact of a liquid and its vapour on the flat surface of an undeformable solid.Their estimate for T for a Lennard-Jones liquid in contact with its vapour is ca. -3 x N. Such a tension is smaller by a factor of 100 than those reported for soap films, but is not much smaller than Buff and Saltsburg's estimate of about (~d,where d is the range of the intermolecular forces, say 5 x lO-"m for argon, or T = N. A similar argument based on a simple analy~is~.'~ in terms of a local excess of free-energy density, suggests that T is usually negative and again of order magnitude -ud, where d is the thickness of an interface, or the correlation range in a dense liquid, or the range of the intermolecular forces, since these three lengths are all of the same order of magnitude. Indeed this estimate, IT^= (~dwhere d is the range of the intermolecular forces, is probably a general one, with ITI=~O-'~Nfor 'simple' systems such as argon and pure water, and as large as 10p9Nfor systems such as soap films in which there are forces of lengths large compared to the intermolecular ~eparation.'~ These considerations of the size of T and of the relevant scales of length raise doubts about the usefulness of expressions for the change of T with R.The formal status of equations such as eqn (4.30) is not in doubt since classical thermodynamics makes no distinction between r and (T and so tells us nothing of the size of the length (T/(T). If, however, the experimental and statistical mechanical evidence suggests that (T/R)is of order ((TSIR),then it is in fact an order of magnitude smaller than (T,cf.eqn (2.13), and its correction term (w/R2)is only of order a(S/R)'. It has been shown elsewhere4,' that such terms are not only small, but are also incapable of consistent and satisfactory definition in terms of the molecular correlation functions. I thank Dr J. R. Henderson for useful discussions on the subject of this paper. G. N. Antonow, J. Chim. Phys., 1907, 5, 372' J. W. Gibbs, The Collected Works (Longmans, Green, New York, 19281, vol. 1, p. 219 et seq.; R. C. Tolman, J. Chem. Phys., 1949, 17,333; F. 0.Koenig, J. Chem. Phys., 1950,18, 449; F. P. Buff, J. Chem. Phys., 1951, 19, 1591; T. I,. Hill, J. Phys. Chem., 1952, 56, 526; S. Kondo, J. Chem. Phys., 1956, 25, 662. 90 THERMODYNAMICS OF A LIQUID LENS J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity (University Press, Oxford, 19821, sections 2.4,4.8 and 8.6. S. J. Hemingway, J. R. Henderson and J. S. Rowlinson, Faraday Symp. Chem. SOC., 1981, 16, 33.' P. Schofield and J. R. Henderson, Proc. R. SOC.London, Ser. A, 1982, 379, 231 ; J. R. Henderson and P. Schofield, Proc. R. SOC. London, Ser. A, 1982, 380, 211. D. Platikanov, M. Nedyalkov and A. Scheludko,J. Coll. Interf. Sci., 1980,75,612;D. Platikanov, M. Nedyalkov and V. Nasteva, J. Colloid Interface Sci., 1980, 75, 620. L. R. Fisher and J. N. Israelachvili, Nature (London), 1979, 277, 548; Chem. Phys. Lett., 1980, 76, 325. F. P. Buff and H. Saltsburg, J. Chem. Phys., 1957, 26, 23. L. Boruvka and A. W. Neumann, J. Chem. Phys., 1977, 66, 5464. 10 G. Navascuis and P. Tarazona, Chem. Phys. Letf., 1981, 82, 586. 11 P. Tarazona and G. Navascuts, J. Chem. Phys., 1981,75, 3114. 12 J. Kerins and B. Widom, J.Chem. Phys., 1982,77, 2061. 13 A. Vrij, J. G. H. Joosten and H. M. Fijnaut, Adv. Chem. Phys., 1981,48, 329. (PAPER 2/822)

ISSN:0300-9238    

DOI:10.1039/F29837900077

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

J. Chern. SOC.,Faraday Trans. 2, 1983,79, 91-109 Motion of Flocs of Two or Three Interacting Colloidal Particles in a Hydrodynamic Medium BY JOHN BACON, ERICDICKINSON”AND ROGER PARKER Procter Department of Food Science, University of Leeds, Leeds LS2 9JT AND NICHOLASANASTASIOUAND MOTI LAL* Unilever Research, Port Sunlight Laboratory, Bebington, Wirral, Merseyside L62 4XN Received 18th May, 1982 Doublets and triplets of secondary-minimum flocculated, DLVO-type colloidal particles have been simulated by the computational technique of Brownian dynamics, and aggregate lifetimes have been determined as a function of well depth. It is found that a pair of particles may “jump” out of a secondary minimum of depth 10 kT within a matter of seconds. Flocs of DLVO-type particles typically “dissociate” on a timescale shorter than that for floc rotational diffusion, and so a harmonic-dimer potential has been used to model the rotational and translational diffusion of a stable doublet. For particles that can approach closely (within, say, 5% of a particle radius), it is necessary to use exact interparticle hydrodynamic expressions rather than the approximate Oseen form.With triplets there is an increasing tendency towards linear “dissociation” as the well depth increases. The simulated doublets are consider- ably less stable than the real doublets of polystyrene latex particles which they are meant roughly to represent. Inevitable ambiguities in defining words like “dissociation” and “aggregate” are discussed. There is increasing interest in applying the methods of statistical mechanics to systems of interacting colloidal particles.Attention in recent years has been directed mainly towards structure and phase equilibria,’ but it is clear that many of the important problems in colloid science are kinetic in nature. Under a particular set of experimental conditions, a common question is: how fast do particles aggregate and what is the probability of their being redispersed? As the initial stage in a computational study of time-dependent behaviour in concentrated dispersions, this paper reports results for two and three particles interacting with pair potentials of the DLVO (Derjaguin-Landau-Verwey-Overbeek) form. A theoretical treatment of the dynamics of colloidal particles must properly take account of the motion of the dispersion medium through what is usually called the “hydrodynamic interaction”. In the past, the hydrodynamic interaction _has been included in trajectory analysis,* in Smoluchowski coagulation theory,” in “suspension mechanics” through studies of the concentration dependence of diff~sion,~viscosity5 and sedimentation,6 and in statistical mechanics through the study of diffusion via the dynamic structure factor for homogeneous systems of interacting particle^.^ Previous computer simulations’ of the dynamics of colloidal systems have involved homogeneous dispersions of particles which, while strongly interacting, are hydrodynamically dilute, thus not requiring the inclusion of hydro-dynamic interactions.Here we consider systems that are strongly coupled through both hydrodynamic and non-hydrodynamic interactions. At any instant in such a system, a colloidal particle can be hagined as being influenced simultaneously by several types of force: (i) hydrodynamic forces which depend on the positions of 91 BROWNIAN DYNAMICS OF FLOCS nearby particles moving with finite relative velocities, (ii) non-hydrodynamic, veloc- ity-independent, interparticle forces (dispersion, "steric", electrostatic etc.), (iii) forces associated with external fields (electrical, gravitational etc.) and (iv) Brownian forces from fluctuating interaction with the solvent molecules (and ions) of the dispersion medium. The concept of a hydrodynamic interaction between particles implies that the dispersion medium can be treated as a continuum.In reality, of course, the continuum model breaks down at extremely close particle separations, in which case the solvent structure and molecular inhomogeneities of the surface should be taken into account. If, as is usual, the dispersion medium is an electrolyte solution, there is an additional Stokes drag (normally <0.1% according to theory') due to ions in the electrical double layer (the primary electroviscous effect). This additional viscous drag may itself be affected by the proximity of other charged particles (this is in addition to the influence of the secondary electroviscous effect). It will be assumed here that colloidal particles are rigid, impermeable spheres, and that non-slip boundary conditions are applicable. The hydrodynamic coupling between two rigid spheres in an unbounded medium is known e~actly,~ but no formal solution is available for three or more spheres. Because, therefore, it can in principle be calculated exactly (i.e.within the statistical uncertainties), the motion of the interacting doublet deserves special attention.It will be assumed that Brownian forces are random in space and time (i.e. the diffusion is a gaussian process) and that external forces are absent (they can easily be included later). In simulating a triplet of interacting particles, we shall additionally assume that hydrodynamic interactions and potentials of mean force are pairwise additive.Pairwise additivity of DLVO potentials is likely to be a valid assumption for large colloidal particles at most ionic strengths of practical interest (Ka >> 1). The computations reported here have a twofold purpose: (a)to investigate how stability and diffusional behaviour of flocs of spherical particles are affected by the depth of the secondary minimum and the assumed form of the hydrodynamic interactions; (6)to compare the computed aggregate lifetimes with those measured by direct optical microscopic observation of polystyrene latex particles." SIMULATION MODEL AND PROCEDURE HYDRODYNAMIC INTERACTION BETWEEN TWO SPHERES At low Reynolds number, the velocities u1 and v2 of two spheres acted upon by external forces F1and F2 can be written as Vl=bllFl+b12F2 (1) v2= b21F1+b22F2 (2) where the torque on each sphere is taken as zero.For identical spheres of radius a suspended in a Newtonian medium of shear viscosity q,the coefficients bI1,b12, bz1and b22in eqn (1)and (2) are defined by +Bij(p)[l-(r12rl2/r?~)I}/6~~abij ={Aij(~)(r12r12/r:2) (i, j = 1,2) (3) where r12is the vector from the centre of sphere 1 to the centre of sphere 2 and I is the unit tensor. The mobility tensor bij contains four independent coefficients (All,A12, B11 and BIZ),each of which depends on the reduced separation P =r12/a (4) J. BACON, E. DICKINSON, R. PARKER, N. ANASTASIOU AND M. LAL 93 where r12 = Ir121. Relative translational motion along the line of centres is deter- mined by the quantity AA (PI = A lib> -A 12w (5) and rotation about the centre of friction by Numerical values of the hydrodynamic coefficients have been tabulated by Bat~helor.~ The simple Oseen expression for the mobility tensor” is obtained by taking only the leading terms in the asymptotic expansions valid for A 1 1 = 1-3.75p-4 + 0( p ‘) (8) Although satisfactory when particles are far apart, the Oseen approximation can be very poor indeed at close separations.This is illustrated in fig. 1,where exact and Oseen forms for AA(p)are plotted in the range 2 sp s2.16. As the two spheres touch (p-+ 2), the exact function vanishes, whereas the Oseen function approaches a finite value of i.At close separations, then, the Oseen mobility tensor grossly overestimates the tendency of two spheres to move towards (or away from) each other along the line of centres. (For p < 2.01, the Oseen expression is wrong by more than a factor of lo!) The Oseen result for AB(p), on the other hand, is reasonably well behaved right down to very close separations.So we expect the rotational motion of aggregates to be less sensitive, in general, to the choice of hydrodynamic approximation than is the relative translational motion. POTENTIALS OF MEAN FORCE For an electrostatic colloid, the interaction energy U(r12) between two particles a distance r12 apart is conventionally written as u(r12)= UR(r12) + uA(r12) (12) where URrefers to electrostatic repulsion and UArefers to van der Waals attraction.In the doublet simulations reported here, the DLVO functional forms for uR(r12) and UA(r12)are chosen to be compatible, as far as possible, with expected potentials of mean force between the polystyrene particles studied microscopically by Cornell et a1.l’ (“latex B”). The repulsive potential for identical particles (Ka >> 1)is taken asI2 UR(r12) = 2.rr&,&oa$iIn [I +exp (-KS12)] (13) where cc/o is the surface potential, F, is the relative dielectric constant of the BROWNIAN DYNAMICS OF FLOCS 0.3 0.2 bA 0.1 02.o 2.05 2.1 2.l5 FIG. 1.-Effect of hydrodynamic approximation on relative translational motion of two spheres. The quantity AA defined by eqn (5) is plotted against the reduced separation p: E, exact hydrodynamics; F,Felderhof approximation ;16 0,Oseen approximation. continuous phase, c0 is the absolute permittivity of free space and s12 = r12 -2a (14) is the surface-to-surface separation.For a 1: 1electrolyte at concentration c and temperature T,the parameter K is defined by K~ = 2e2~N/~,~okT (15) where e is the electronic charge, k is Boltzmann’s constant and N is Avogadro’s constant. The retarded attractive potential is taken as13 UA(r12)= -&a/[l2~12(1+ 1.77pdl (16) where AH is the effective Hamaker constant and p12is defined by p12 = 2‘rrS12/AL (17) where hLis the London wavelength. Eqn (16) is valid13 for s12<<CI and O<p12< 2, which is quite satisfactory for the range of separations with which we are concerned here.The assumed parametric values are: a = 1pm, cr= 80 (approximately the value for water), AH = 5 x J (polystyrene in water), T = 300 K, $0 = 25 mV (taken to be independent of c), q = 0.865 mPas (water) and AIL= 0.1 pm. Fig. 2(a)shows a plot of U(r12)against the reduced separation p for three different ionic strengths: potential (1)with c = 10 mol mP3,potential (2) with c = 50 md m-3 and potential (3) with c = 120 mol m ’. Listed in table 1 are the positions and depths, rmin and of the secondary minima. We have also studied a model dimer with a harmonic potential of the form ~(~12)= QC (r12 -rE12 -EE (18) where rEis the “equilibrium” centre-to-centre separation, eE is the well depth and J. BACON. E. DICKINSON, R. PARKER, N.ANASTASIOU AND M. LdAL 95 20 10 go -10 FIG.2.-Pair potentials of mean force used in Brownian dynamics simulations of (a)doublets and (b) triplets: (1)-(6) DLVO-type potentials; (---) harmonic potential. The energy U (divided by kT) is plotted against the reduced separation p. (See text for further details.) CY is a steepness parameter. The numerical values of the parameters in eqn (18) are derived from a Taylor series expansion about the bottom of the secondary minimum of DLVO potential (2) in fig, 2(a). The fitted values are TE = 2.008 pm, EE = 4.68kT and a = 1.016X J mP2, BROWNIAN DYNAMICS OF FLOCS With the potentials used in the triplet simulations, the secondary minima are located much further away from the particle surfaces [see fig.2(b)]. Mathematically, this is achieved with UR(rf2)given by eqn (13) with a = 0.5 pm, E~=80, $0 = 30.5 mV and KU = 20 [c = 0.15 mol mP3 and T = 300 K in eqn (15)], and an attrac- tive potential of the formf4 ZLdh2) = -(AH/12)“ -2)Pl -l+ (P -+2 In [(P -2)P/(P -u”1) (19) Fig. 2(b) shows U(rI2)plotted against p for three different values of the Hamaker constant: potential (4) with AH= 0.83 x J (crnin= 0.8kT),potential (5) with AH= 1.66x J (tzmin= 2.4kT) and potential (6) with AH= 2.49 X lopf9J (cmin= 4.0kT). BROWNIAN-DYNAMICS ALGORITHM The simulation technique of Ermak and McCammon” is used to generate configuration-space trajectories composed of successive displacements of time-step At. The Brownian-dynamics method resembles the more familiar deterministic molecular-dynamics algorithm used to simulate simple liquids.The main difference is that the usual set of 3N Newtonian equations is replaced by a set of 3N Langevin equations. In the nomenclature of Ermak and McCammon l5 the equation of motion of N particles is then r,-ry = C(dDy,/dr,)At +Chy,FyAt +R,(Dy,,At) (1si, j s3N) (20) I 1 where rr represents a single position coordinate (in the x, y or z direction), the superscript ’refers to the beginning of a time-step and F, is the magnitude of a systematic force arising from interparticle and external field interactions in direction j. The algorithm stipulates that At be considerably larger than the decay time of the particle velocity autocorrelation function.The symmetric, configuration- dependent diffusion matrix D,, is related to the mobility matrix b,, by the Einstein equation: D,,= kTb,, (i,j = 1,3N). (21) The matrix b,, (i, j = 1,3N) in eqn (21) is constructed from the set of mobility tensors b,, (i, j = 1,2) [see eqn (3)]for all pairs of particles in the system. The quantity R,is a fluctuating quantity which depends on the diffusion tensor D,, and the time-step At. It is convenient to regard the right-hand side of eqn (20)as consisting of three distinct terms: a “gradient” term, a “hydrodynamic” term and a “stochastic” term. At the level of the Oseen approximation [eqn (7)] the gradient term vanishes trivially. However, with proper hydrodynamics the force associated with the gradient term is of short range and always repulsive.The role of the hydrodynamic term in eqn (20) is to couple the mobility to the non-hydrodynamic interparticle forces and the external forces (if any). In turn, the stochastic term is coupled to the diffusion tensor and therefore to the hydrodynamic interactions. The displace- ment R,is a random quantity with a gaussian multivariate distribution defined by (Ri(At))= 0 (22) (Ri(At)Rj (At)) = 2DpjAt (23) where ( -* * ) denotes the average value. J. BACON, E. DICKINSON, R. PARKER, N. ANASTASIOU AND M. LAL 97 TABLE1.-CHARACTERISTIC PARAMETERS OF DLVO-TYPE POTENTIALS USED IN DOUBLET SIMULATIONS [SEE FIG. 2(a)] 1 10 0.87 2.024 2.0025 2 50 4.68 2.008 2.0015 3 120 10.67 2.005 2.0010 CHOICE OF TIME-STEP The efficiency of a computer simulation relies on taking as large a time-step as numerical accuracy permits.Here, At must be small enough for F,, Dijand (dDij/dr,) to be essentially constant during a time interval. With DLVO-type interactions, it is the steepness of the short-range repulsive region that in practice determines the maximum time-step. For potentials (1)-(3) in fig. 2(a) At must be < 1ps to avoid the possibility of particles being “pushed” over the primary maximum? during a run of, say, lo6 cycles. Such a short time-step is, however, unsatisfactory: it gives a real simulation time of <1s for the run, which may be from one to three orders of magnitude shorter than that of interest to the colloid scientist.To circumvent this problem in our study of flocculated doublets with potentials (1)-(3), we have used longer time-steps (up to 50 ps), whilst simultaneously preventing spurious coagulation into the primary minimum by artificially holding the repulsive force at a constant value for r <rc. For completeness, the numerical values taken for rc are listed in table 1, but it was found that results were essentially independent of the values chosen. The above problem does not arise with softer potentials (4)-(6) [see fig. 2(b)] used in connection with the triplet simulations. [With two particles, spurious coagulation could alternatively be overcome by making At a function of the pair separation (“multiple time-step” method). For aggregates of three (or more) particles, with some pairs close and others not, it is doubtful whether the extra programming complexity of multiple time-steps would be worthwhile.] For the same number of time-steps, doublet runs involving the Oseen approxi- mation were faster (ca.20°/0j than those employing exact hydrodynamics. This is because the simple analytic Oseen expressions can be evaluated explicitly at the appropriate point in the computer program, whereas the exact hydrodynamic coefficients, after fitting to polynomials with the right functional forms, must be extracted from a table. Hence, for particles that are prevented electrostatically from approaching closely [e.g. with potentials like those in fig. 2(b)], there is definitely something to be said, in terms of computer time and storage, for using the Oseen approximation.(There seems little computational advantage in anything intermediate between the Oseen and exact forms; for example, the approximation of Felderhof16 illustrated in fig. 1.) That having been said, for particles which can approach closely (say, p <2;2), which includes most particles in real colloidal dispersions, the Oseen tensor’s small advantage in efficiency per unit computational time-step is far outweighed by the fact that it overestimates the relative translational mobility. This makes it necessary to use shorter time-steps to get results of :With a gaussian distribution driving the stochastic term there is, of course, a non-zero chance of “overlap” with any value of At, however small.This being the case, we note that so long as the time-scale over which results are required is large compared with the time-step, Rimay in fact be chosen randomly from any distribution (central limit theorem). Another way of getting round the overlap problem, then, is with a distribution having a finite supremum. 98 BROWNIAN DYNAMICS OF FLOCS equivalent numerical precision, results that are in any case in error by an indetermin- ate amount. Simulations of doublets were run on the CDC 7600 machine in Manchester; those of triplets on the Amdahl V7 machine in Leeds. The computations are expensive on computer time, a factor which could put limitations on the types of simulation feasible for flocs containing a large number of particles.For the harmonic dimer simulation reported here, for instance, 3 x lo7 cycles consumed 1h C.P.U. time (10 min real time of particle movement for a typical time-step of 20 ps.) With At = 20 ps, a harmonic dimer with an Oseen hydrodynamic interaction spends ca. 1time-step in every 2500 at separations less than p = 2.0! This particle overlap is obviously unphysical (as far as relative translational motion is concerned), but it does not cause the program to fail. More importantly, it seems not to lead to any artefacts in the rotational results, since translation and rotation are effectively uncoupled. To obtain reliable medium-time orientational correlation functions, long runs with relatively large values of At are needed in order to sample enough configurational space.In simulating a harmonic dimer with exact hydrodynamics, it was necessary to take the radial hydrodynamic force to be constant at p <2.003. This is because, with a relatively large value of At (ca. 20 ps), once a particle is “pushed inside” another (p <2), the exact hydrodynamic term in eqn (21) can never overcome the effect of the stochastic term [AA --+ 0 as p --+ 2 (see fig. l)]. RESULTS The kinetics of deflocculation of pairs of particles (diameter 2 pm) with potentials (1)-(3) [see fig. 2(a)]have been studied. Starting at the bottom of the secondary minimum (rI2= rmin),a doublet was deemed to have “dissociated” when the separ- ation exceeded p = 2.2 (the net attractive energy here is less than a few per cent of kT).Distributions of lifetimes are built up by repeating the simulations a large number of times with different sets of random numbers. The mean lifetime 7 is shown in fig. 3 as a function of time-step, ionic strength and hydrodynamic approxi- mation. With exact hydrodynamics, the mean lifetime is independent of time-step for At s20 ps. With Oseen hydrodynamics, however, the position is less satisfac- tory: for c = 10, 50 and 120 mol m-3, 7 is independent of time-step below 20, 5 and (probably) 1ps, respectively. By comparing fig. 4 and 5, we see the influence of the hydrodynamic approxima- tion on the doublet lifetime frequency distribution f(7) at c = 10 mol m-3. The two normalised distributions have similar general shape, but the mean lifetime given by the Oseen approximation (7 = 0.131s) is much less than that given by exact hydrodynamics (7=0.258s).In both cases there is a maximum in f(7) at 7 = 0.4 7. By comparing fig. 4 and 6, we see the influence of ionic strength on the doublet lifetime frequency distribution. As expected, the deeper the secondary minimum, the greater is the dimer stability. Apart from the scale difference along the abscissa, the forms of the normalised distributions in fig. 4 and 6 are similar. (Statistics are poorer in fig. 6 owing to the lower number of “dissociations”.) At an ionic strength of 50 mol mP3, the mean lifetime is 1.49s for exact hydrodynamics and 0.56s for Oseen hydrodynamics. At an ionic strength of 120 mol mP3, the mean doublet lifetime is 20 f5 s (based on just two “dissociations”!).It is difficult to get reliable data on the translation and rotation of DLVO doublets, since, at essentially all ionic strengths, they tend to “dissociate” on a time-scale not greater than that for doublet rotation. With this in mind, the J. BACON, E. DICKINSON, R. PARKER, N. ANASTASIOU AND M. LAL 99 \ -A A-UY -&-A-A, ‘, ---E It--v T---x---0 V---O-‘O ----L------------A---3 * -------_---_--0--I I I I 0.101 0 0.25 0.5 0.75 1.0 71s FIG. 4.-Doublet lifetime frequency distribution for 2 Fm diameter particles with DLVO pair potential (1)and exact hydrodynamics. The normalised function f(7)is plotted against the lifetime 7 (from 543 “dissociations”).IS 0 0.25 0.5 0.75 71s FIG.5.-Doublet lifetime frequency distribution for 2 Fm diameter particles with DLVO pair potential (1)and Oseen hydrodynamics. The normalised function f(7)is plotted against the lifetime 7 (from 1300 “dissociations”). BROWNIAN DYNAMICS OF FLOCS -0.10 mm I I 0 4 6 T/S FIG.6.-Doublet lifetime frequency distribution for 2 pm diameter particles with DLVO pair potential (2) and exact hydrodynamics. The normalised function f(~)is plotted against the lifetime 7 (from 185 "dissociations"). 1.0 FIG.7.-Rotational correlation function for harmonic dimer. The function A,([)is defined by eqn (24): (-) exact hydrodynamics; (---) Oseen hydrodynamics. Results are based on runs lasting 13; min (4x lo7 cycles) and 20 min (5.5 x lo7 cycles) for exact and Oseen hydrodynamics, respectively.harmonic dimer was studied as an example of a genuinely stable doublet. Fig. 7 shows the rotational correlation function Ar(t) = (COS 0 (t))= (312(0) j12(t)) (24) where 0 represents the change in angle of the unit vector tI2during time t. Plots of InA, against t were found to be linear within 1% in the range 1aAr30.5 independent of the assumed form of hydrodynamic interaction and the time-step (650ps). Values of the rotational diffusion coefficient D, defined by A,(t)= exp (-2D,t) (25) are given in table 2. We see that the Oseen approximation overestimates rotational mobility by ca. 30%. For an aggregate, there are two translational diffusion coefficients which may be of interest: the self-diffusion coefficient of an individual particle, D,, and the diffusion coefficient of the aggregate as a whole, D,.If the aggregate is genuinely stable, the two diffusion coefficients will become equal in the limit t -+ a;but, at finite times, we would expect to observe D,>D, due to additional mobility within the aggregate. With just two particles, of course, this extra mobility is simply manifest in the relative rotation, i.e. D,-+D, as A,+O. The self-diffusion J. BACON, E. DICKINSON, R. PARKER, N. ANASTASIOU AND M. LAL 101 TABLE 2.-DIFFUSION COEFFICIENTS FOR SIMULATED HARMONIC DIMER hydrodynamicapproximation computed theory[eqn W)1 Ds m2s-' Da /10~13m2s Oseen 0.073 f0.009 0.0789 2.40 1.90 exact 0.058 f0.006 0.0619 2.37 1.84 TABLE3.-MEAN p AND,STANDARD DEVIATION u OF DISTRIBUTION OF SEPARATIONS FOR HARMONIC DIMER hydrodynamic approximation Am P 10' 0 Oseen 5 2.008 003 1.55, Oseen 20 2.007 997 2.34, exact 20 2.007 99, 1.45, exact 50 2.007 976 1.50, (Boltzmann) 2.008 1.427 coefficient of particle i in the simulation is given by ~s = ([ri(0)-ri (tj12)/6t (26) and plots of mean-square displacement against time for the harmonic dimer were found to be linear (within *lo/o) for O~t/s~5,both for exact and Oseen hydro- dynamics. Values of D, given in table 2 lie ca.6% below the value of the Stokes single-particle diffusion coefficient (2.54x m2s-l). The values of D, com-puted over the same time-scale (ts5 s) lie ca.25% below the Stokes value. An additional check of the numerical scheme for a stable doublet is to see how close to a Boltzmann distribution is the distribution of pair separations in the limit t +a.The harmonic dimer potential [eqn (18)] gives a theoretical distribution P(p)of the normal form, i.e. P(p>= (P/d1/2exp [-P(P -d21 (27) where PE = k/a (28) p =cra2/kT. (29) The standard deviation cr is given by cr = (2p)y2. (30) The theoretical Boltzmann distribution for the harmonic dimer potential illustrated in fig. 2(a)has a standard deviation (TTB = 1.427 x Table 3 lists some computed values of cr for comparison. With exact hydrodynamics, the computed value is 5% larger than (TTB for At = 50 ps, and just 2% larger than CTTB for At = 20 ps.Oseen hydrodynamics, as expected, gives less good agreement: the errors are 65% and 9% for time-steps of 20 and 5 ps, respectively. These time-averaged dimer results tell BROWNIAN DYNAMICS OF FLOCS 0.1 -h t-d i -0-1 60 120 180 dInaxl0 F1~.8.-Statistics of triplet “dissociations” for 1pm diameter particles with DLVO pair potential (4) and pairwise-additive Oseen hydrodynamics. (a)The normalised lifetime distribution function f(7)is plotted against the lifetime 7. The last column of the histogram includes all lifetimes >1s. +,,,is plotted against the angle f(q5max)normalised angular distribution function (b)The (from 287 “dissoci-ations” with 7 G 1s). the same story as the mean lifetimes plotted in fig.3; i.e. with exact hydrodynamics we get reliable results with a time-step of ca. 20ps, whereas with Oseen hydro- dynamics a much smaller increment is needed for simulating radial motion. As a final comment on table 3, we note that the mean separation p of the harmonic dimer is, perhaps not surprisingly, insensitive both to the time-step and the hydro- dynamic approximation. The rate of “dissociation” of triplets of particles of diameter 1pm with ’pair potentials (4)-(6) [see fig. 2(6)] has been studied at the level of the Oseen approxi- mation. Particles were started equidistant from one another (r12 =r13 =r23 =rmin) with the system at its lowest potential energy. A simulated triplet was said to have “dissociated” into monomer +dimer when two of the distances r12, r13 or r23 became >1.5 Fm (i.e.3a).Fig. 8(a),9(a)and 10(a)show normalised lifetime distributions J. BACON. E. DICKINSON, R. PARKER, N. ANASTASIOU AND M. LAL 103 1 I i 0.1 h b< 0 0 1 0.2 - 0.1 -8 4 I h i 0 4 60 120 180 4maxlo FIG. 9.-Statistics of triplet “dissociations~’for 1pm diameter particles with DLVO pair potential (5) and pairwise-additive Oseen hydrodynamics. (a)The normalised lifetime distribution function f(~)is plotted against the lifetime T. The last column of the histogram includes all lifetimes >1 s. (b)The normalised angular distribution function f(4max)is plotted against the angle &,,, (from 193 “dissoci-ations” with T s 1s).f(~)for triplets that “dissociate” within 1s. Again, as expected, the stability of the floc increases strongly as the depth of the secondary minimum is increased, Whereas 99% of particles with pair potential (4)“dissociate” within 1s, only about two in every five particles with pair potential (6) “dissociate” within the same time. With aggregates of more than two particles, one is also interested in the changing shape of aggregates during “dissociation”. Some indication of this for triplets is shown in fig. 8(b), 9(b)and 10(b). The normalised frequency distribution f(4)max) is plotted as a function of the largest angle between pairs of unit vectors taken from $12, $13 and $23 (q5max= 60”corresponds to the initial configuration; 4max = 180” represents the exactly linear arrangement).Each value of 4maxrefers to a configur-ation at the instant of “dissociation” (as defined above), and only those “dissoci-ations’’ occurring within 1s are included in the statistical analysis. We see that for 104 BROWNIAN DYNAMICS OF FI-OCS 0.2 -0.1 b< 0 0 0.5 1 T/S 0.2 I 60 120 180 drnaxl’ FIG. 10.-Statistics of triplet “dissociations” for 1Fm diameter particles with DLVO pair potential (6) and pairwise-additive Oseen hydrodynamics. (a)The normalised lifetime distribution function f(~)is plotted against the lifetime 7. The last column of the histogram includes all lifetimes >1 s. (b) The normalised angular distributions function f(q!~,,,) is plotted against the angle d,,, (from 217 “dissoci-ations” with T G 1s).each of the potentials of mean force the most probable configuration corresponds to 4,,, =:70”. The mean values of 4maxfor potentials (4),(5)and (6)are respectively (76f2)”,(82f2)”and (91f3)”,which means that as the well depth becomes greater there is a greater probability of open configurations. With potential (6) an appreci- able number of triplets take up linear (or approximately linear) configurations prior to “dissociation”. That the triplet “dissociations” summarized in fig. 8-10 can in no way be regarded as final is conveniently illustrated by fig. 11. Starting with pair separations at the potential minima, representative sets of separations (r12, r13 and r23) for potentials (4) and (6) are recorded over a time period of 5 s.As expected, the triplet of particles interacting with potential (6) is more tightly bound than the triplet of particles interacting with potential (4). Indeed, the latter triplet is in the J. BACON, E. DICKINSON, R. PARKER, N. ANASTASIOUAND M. LAL 105 6 0 1 2 3 4 5 tls p4 NL 2 FIG. 11.-Particle pair separations in 5 s triplet simulations based on same set of random numbers. The separations r12, r13 and r23 are shown as a function of time t: (---) potential (4);(-D-H-) potential (6). dissociated state (as defined above) for at least half of the simulation period. In order to illustrate graphically the role of the stochastic term in determining the particle dynamics, the two sets of data shown in fig.11were generated using exactly the same set of random numbers. Other triplet simulations have been carried out with particles starting from different positions with respect to each other. The results show similar general features to those discussed above; i.e. irrespective of the criterion of “dissociation”, the average lifetime and the width of the lifetime distribution function both increase with the depth of the secondary minimum. BROWNIAN DYNAMICS OF FI,OCS DISCUSSION The sensitivity of the particle dynamics to the hydrodynamic approximation depends on the sort of motion we are considering. The rotational correlation functions for the stable doublet (see fig. 7)show that rotational mobility is insensitive to interparticle hydrodynamics even when particles are (on average) quite close (p <2.01). Table 2 compares the computed rotational diffusion coefficient for the harmonic dimer with that expected theoretically for a rigid dumbbell D, = AB(p,)kT/3mp3p; (31) where po ( =pE)is the fixed value of the reduced separation. Within the computa- tional uncertainty, the value of R,for the harmonic dimer is the same as that for the rigid dimer with a = 1.0 pm and po= 2.008 pm.This is to be expected. Because of (i) the decoupling of rotation and relative translation, (ii) the effective linearity of D,(p) over the effective width of P(p) (i.e. pE*3u) and (iii) the symmetry of P(p)about p =pE, it is easy to show that we expect D, for the harmonic dimer to be negligibly different from that for the rigid dimer.That this is indeed found is further verification of the numerical procedure. For both types of dimer, the Oseen approximation gives values that are ca. 30% too high. Table 2 shows that D, and D, are fairly insensitive to the hydrodynamic approximation. While rotational motion and absolute translational motion do not depend strongly on interparticle hydrodynamics, the same cannot be said of relative transla -tional motion, especially at close separations. In connection with sedimentation, it has been pointed out l7already that the effect of double-layer repulsion in keeping colloidal surfaces apart reduces considerably the uncertainties that may arise from the nature of interparticle hydrodynamics at short range.With unstable aggregates, whose particle surfaces approach within, say, 5% of the particle radius, two points are relevant in connection with the use of the Oseen approximation, or, for that matter, any approximation (e.g. that of FelderhofI6) which gives AA +0 as p +2 (see fig. 1). First, computed lifetimes may be too low by a factor of 2 or 3 even if At is satisfactory. Secondly, the time-step must be set extremely short for reliable results, especially at high ionic strengths. With the Oseen approximation at c = 120 mol mP3, if the time-step is set at 20 ps, for instance, the (erroneous) computed mean lifetime is very short indeed (see fig. 3). This arises because a large number of doublets last just a few cycles: the stochastic term soon “pushes” the pair into the steeply repulsive region of the primary maximum, which then induces the doublet to “fly apart” in a subsequent single time-step.With exact hydrodynamics, this problem does not arise. The doublet lifetime distributions shown in fig. 4-6 all contain a maximum at T =0.4 7. The low probability of a doublet “dissociating” extremely fast is due to the fact that two particles take a finite time to diffuse apart even in the complete absence of any interparticle forces. The same is obviously true as well for aggregates of three or more particles. Before comparing the simulation results of fig. 2-6 with the experimental observations of Cornell et al.,” we must address ourselves to the question of the practical relevance of the word “dissociation” as used in this paper (passim).In the microscopic work, a lifetime is reportedly determined by taking the time from the point at which a pair of particles is observed to “collide” [i.e. the point at which there is no perceivable separation between them (p =2.5)] and the point at which they are observed to separate again. Since computer time is limited, it is not feasible in a simulation to wait, over and over again, for pairs of widely separated J. BACON, E. DICKINSON, R. PARKER, N. ANASTASIOU AND M. LAL 107 FIG. 12.-Representation of paths making up total trajectory for encounter between two colloidal particles: (a)looping in region I1 from r12 =rexp;(b)traversing region I1 from r12=rexpto r12=r*; (c) traversing region I1 from rt2 =r* to r12=rexp;(d)looping in region I1 from r12 =r*; (e)looping in region I from r12=r*.The potential of mean force is effective only in region I. particles to come together. Neither is it feasible to wait for them to diffuse a long way apart again. It must be recognised that any definition that distinguishes aggregated and unaggregated particles will have some element of arbitrariness about it. Nonethe- less, it does seem a reasonable criterion to assume that two particles are unaggre- gated if the separation r12 exceeds some distance r* at which U(r*)<<kT [say, U(r*)=O.lkT]. With colloidal particles large enough to be observed microscopi- cally, the observed separation at doublet break-up (r12=rexp) will depend on the optical resolution and the orientation of the doublet with respect to the observer.We can postulate two regions within the range 2a r12 rexp: region I (2a r12 r*), where dynamical behaviour is dependent on electrostatic and van der Waals forces, and region I1 (r* sr12srexp), where motion is controlled by hydrodynami- cally-coupled Brownian forces in the absence of any significant contribution from the potential of mean force. The trajectory during a binary encounter may be regarded as consisting of a combination of the five paths (a)-(e)illustrated in fig. 12, with paths (d)and (e)perhaps occurring several times over. We note that only path (e) depends on electrolyte concentration; it is the average time for traversing (half) this path which is most closely related to the simulation lifetimes reported here.In this paper we have distinguished between the words “dimer” and “doublet”. The term “dimer” implies either (a) that two particles are permanently bound together (as with the harmonic dimer) or (6) that the average time for which one particle remains within about a diameter of the other is very much greater than the average time it takes for an isolated particle to diffuse over its diameter. According to this definition, the pairs of DLVO particles simulated here do not form dimers [with the possible exception of those interacting with potential (3)]. A “doublet” is defined to be a pair of particles whose separation is within the range of the interparticle forces, however weak.If the interparticle forces are strong, a “doublet” may also be a “dimer”. In the complete absence of attractive interparticle forces, doublets do not exist at all. 108 BROWNIAN DYNAMICS OF FLOCS The mean doublet lifetimes observed experimentally by Cornell et al.” are in the range 3-10 min depending on ionic strength, compared with 0.2-20 s for the simulated doublets. Part of this discrepancy may be attributable to the change in experimental surface potential with ionic strength, or the difference between the simulation “dissociation” criterion (r12 = r* = 2.2 pm) and the experimental one (r12= rexp= 2.5 pm). But this surely cannot be the reason for such a large dis- crepancy, and so we must look to other factors.One point that should not perhaps be overlooked is the extent to which any observer looking down a microscope can be totally rigorous from the statistical viewpoint. “Dissociations” taking place along, or close to, the direction of observation will not be detected. In addition, it is likely that doublets reassociating rapidly will be excluded inadvertently from the experimental data analysis. This tendeficy to give “the benefit of the doubt” in the direction of the aggregated state, especially for doublets not orientated perpendicular to the direction of observation, might imply an effective value of rexp considerably greater than 2.5 pm. As far as the doublet simulations are concerned, the only assumption reasonably open to question, we believe, is the form of the potential of mean force.What is quite clear from the simulations is that two particles can get out of a well of depth ca. lOkT in a matter of seconds! It is certainly possible that the value which we (and others) have taken for the effective Hamaker constant may be too small. Indeed, with the value assumed, potential (3) lies close to the limiting depth of the secondary minimum: for c >> 120 mol mp3, primary minimum coagulation is possible. One factor that should not be ruled out, despite its almost heretical connotations, is the possibility that the assumed form of DLVO potential between polystyrene latex particles is, in fact, incorrect. That is, latex flocs are perhaps more stable than expected because of the presence of some other attractive interpar- ticle force not included in the present analysis.Results for the triplet “dissociations” are based on the assumption of pairwise additivity of the hydrodynamic forces. This assumption will be poorest for approxi- mately linear configurations, where the middle particle will tend to “shield” the hydrodynamic interaction between the other two. In simulating large flocs or continuous concentrated colloidal systems, more attention will have to be directed towards investigating the sensitivity of the results to this assumption of pairwise additivity. E.D. acknowledges financial support (for J.B.) from the S.E.R.C. R.P. and N.A. acknowledge receipt of S.E.R.C. CASE Studentships.We thank Mrs G. M. Watson (Unilever) for some preliminary programming work on this problem. E. Dickinson, Colloid Science, ed. D. H. Everett (Specialist Periodical Reports, Royal Society of Chemistry, London, 1982), vol. 4, chap. 4. E. B. Vadas, R. G. Cox, H. L. Goldsmith and S. G. Mason, J. Colloid Interface Sci.,1976, 57, 308. L. A. Spielman, J. Colloid Interface Sci., 1970, 33, 562. G. K. Batchelor, J. Fluid Mech., 1976, 74,1. G. K.Batchelor, J. FluidMech., 1970, 41,545. G. K. Batchelor, J. FluidMech., 1972, 52, 245. B. J. Ackerson, J. Chem. Phys., 1978,69, 684. ’ K. J. Gaylor, I. K. Snook and W. van Megen, J. Chem. Phys., 1981, 75,1682. R. W. O’Brien and L. R. White, J. Chem. SOC.,Furaday Trans. 2, 1978, 74,1607. 10 R. M.Cornell, J. W.Goodwin and R. H. Ottewill, J. Colloid Irterfuce Sci., 1979, 71,254. 11 t1. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971), p. 349. J. BACON, E. DICKINSON, R. PARKER, N. ANASTASIOU AND M. LAL 109 12 E. J. W. Verwey and J. Th. G. Overbeek, Theory of Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948), chap. 9. 13 J. H. Schenkel and J. A. Kitchener, Trans. Faraday SOC., 1960,56, 161. 14 H. C. Harnaker, Physica, 1937, 4, 1058. 15 D. L. Ermak and J. A. McCammon, J. Chem. Phys., 1978,69,1352.16 B. U. Felderhof, J. Phys. A, 1978, 11,929. 17 E. Dickinson, J. Colloid Interface Sci., 1980, 73,578. (PAPER 2/827)

ISSN:0300-9238    

DOI:10.1039/F29837900091

出版商:RSC    

年代:1983

数据来源: RSC