Polyelectrolyte-Compression Forces between Spherical DNA Brushes
Kati Kegler, Martin Konieczny, Gustavo Dominguez-Espinoza, Christof Gutsche, Matthias Salomo, Friedrich Kremer, Christos N. Likos
aa r X i v : . [ c ond - m a t . s o f t ] O c t Polyelectrolyte-Compression Forces between Spherical DNA Brushes
Kati Kegler, Martin Konieczny, Gustavo Dominguez-Espinosa, Christof Gutsche, Matthias Salomo, Friedrich Kremer, and Christos N. Likos
2, 3 Institute for Experimental Physics I, University of Leipzig, Lin´eestraße 5, D-04103 Leipzig, Germany Institute for Theoretical Physics II: Soft Matter, Heinrich-Heine-Universit¨at D¨usseldorf,Universit¨atsstraße 1, D-40225 D¨usseldorf, Germany The Erwin Schr¨odinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria (Dated: October 23, 2018)Optical tweezers are employed to measure the forces of interaction within a single pair of DNA-grafted colloids in dependence of the molecular weight of the DNA-chains, and the concentrationand valence of the surrounding ionic medium. The resulting forces are short-range and set in as thesurface-to-surface distance between the colloidal cores reaches the value of the brush height. Themeasured force-distance dependence is analyzed by means of a theoretical treatment based on thecompression of the chains on the surface of the opposite-lying colloid. Quantitative agreement withthe experiment is obtained for all parameter combinations.
PACS numbers: 82.35.Rs, 82.70.Dd, 87.80.Cc
Surface treatment of colloidal particles and the ensu-ing manipulation and control of their interaction prop-erties is a topic of high and lasting interest, on thegrounds of both technological relevance and fundamen-tal importance. On the first count, the main issue per-tains to the fact that surface treatment is necessary toachieve colloidal stabilization by inducing thereby a re-pulsive force that acts against the ubiquitous dispersionattractions between the colloids. Charge stabilizationand steric stabilization, the latter being caused by graftedpolymer chains, are the two most common mechanisms,whereas grafting of polyelectrolyte (PE) chains on a col-loid provides a natural combination of both and resultsto an electrosteric repulsion. On the second count, sur-face treatment by polymer grafting provides the possibil-ity to tune the effective colloid interaction by ‘dressing’the hard sphere potential with a soft tail, whose range,strength and overall functional form can be controlled bychanging the properties of the polymer brush, e.g., itsgrafting density, height or charge. Systems interactingby a combination of a hard sphere potential and a sub-sequent short-range repulsion show a tremendous varietyin their equilibrium [1, 2, 3, 4] and dynamical [5, 6, 7]properties.Considerable work has been carried out in the study ofthe so-called osmotic
PE-brushes [8, 9, 10], which resultfor high surface grafting densities and are characterizedby the fact that they spherically condense the vast ma-jority of the counterions released by the chains. These,in turn, bring about an entropic effective force betweenthe brushes, which has been quantitatively analyzed forPE-brushes [11] and stars [12]. On the other hand, lit-tle is known for the opposite case of low surface graft-ing density, for which the theoretical considerations thatlead to the interaction between osmotic brushes breakdown. In this Letter, we investigate by a combination ofsensitive and accurate experiments and theoretical anal- ysis the effective forces between spherical DNA brushesand establish a novel mechanism of interaction betweenthose, which results from the mutual compression of PE-chains of the colloids against the surface of each other.The quantitative characteristics of the resulting forcesare vastly different from those between osmotic brushes.The experimental investigation was based on the mea-surement of the force-distance dependence between thebrushes employing optical tweezers. Optical tweezers of-fer unprecedented accuracy down to the pN-domain and3 nm in measuring forces and position, respectively. Bymonitoring the force-distance dependencies between twografted colloids it is possible to know how the differentphysicochemical properties (molecular weight, graftingdensity, ionic strength of the surrounding medium) af-fect the effective interaction between the grafted colloids.The force F ( D ) and the surface-to-surface separation D between two identical, negatively charged DNA-graftedcolloids is measured using optical tweezers in a identicalset-up as used in Ref. [13].We employed colloidal particles with a hard core ra-dius R c = 1100 nm, on which DNA-strands with variousnumbers of base pairs (bp) and grafting densities σ werechemically anchored. The brush has only a slight de-viation from planarity, which allows us to relate also toknown facts from planar PE-brushes in what follows. Theforce separation dependence between DNA-grafted col-loids with σ = 8 . · − chains / nm and varying molec-ular weights of the chains is shown in Fig. 1. By usingshorter and shorter DNA-segments, the force displays agradual transition from a soft to a hard sphere potential.The theoretical curves based on the model explained be-low are also incorporated. A reliable estimate for thebrush thickness is deduced by determining the interac-tion length λ F at forces of 2 pN, 4 pN, and 6 pN, see in-set of Fig. 1. For molecular weights between 500 bp and1000 bp, a linear scaling is observed, in accordance with D [nm]0246810 F [ p N ] N = 250 N = 500 N = 750 N =1000
250 500 750 1000 N λ F [ n m ] F =2 pN F =4 pN F =6 pN FIG. 1: F ( D ) curves between DNA-grafted colloids inbuffered solution (10 mM C H NO , pH 8.5) for graftingdensity σ = 8 . · − nm − and various bp-number N , in-dicated in the legend. Symbols: experiments; lines: theoryaccording to Eq. (2), with the values for L and Z eff given inTable I. The inset illustrates the dependence of the interac-tion length λ F on N for three different values of the force. Ref. [9]. Deviations at chain length 250 bp are attributedto the relative increase of the interaction forces betweenuncoated surfaces.In Fig. 2 we show the F ( D ) dependence on the concen-tration and valency of added salt, using the ionic strength I = ( P i c i z i ) / c i and valency z i . Inorder to eliminate possible uncertainties due to variationsamong the colloids, the experiments for the data shownin Fig. 2 were carried out with one single pair of colloidsfor which the solvent is exchanged. With increasing saltconcentration the force-separation dependence becomesshorter-range, reflecting the transition from an osmoticto a salted brush [13] and the concomitant shrinkage ofthe latter. The trends are the same independently of thecounterion valency (NaCl, CaCl , and LaCl ). As for thesalt-free case, we obtain an estimate for the brush thick-ness as the interaction length λ F at the force of 2 pN;results for this quantity are shown in Fig. 3. The slopeof the brush thickness versus ionic strength is close to0 . ± .
05, in good agreement with the scaling law [8].The transition from the osmotic to the salted brush takesplace when the external salt content equals the counte-rion concentration inside the brush. A contact betweenthe solid surfaces of the particles is not observed at lowsalt concentration ( < D [nm]0246810 F [ p N ] I = 0.0 mM I = 0.1 mM I = 3.3 mM I = 10.0 mM I =100.0 mM(a) NaCl0 50 100 150 200 250 300 D [nm]0246810 F [ p N ] I = 0.03 mM I = 0.3 mM I = 3.0 mM I = 9.9 mM I =30.0 mM(b) CaCl D [nm]0246810 F [ p N ] I =0.0 mM I =0.6 mM I =6.0 mM(c) LaCl FIG. 2: F ( D ) dependence for various ionic strengths I andtypes of salt. (a) NaCl; (b) CaCl ; (c) LaCl . Here, N = 1000bp and σ = 8 . · − nm − . Symbols: experiment; lines:theory according to Eq. (2), L and Z eff are given in Table I. counterions that are spherically trapped within the star’scorona. This consideration has been extended to spher-ical brushes, which possess a rigid, colloidal core. Ananalytical expression for the entropic effective interac-tion, V en ( D ), between two brushes at surface-to-surfacedistance D has been derived, reading [11]: βV en ( D ) = N trap (cid:26) D + 2 R c RK ln (cid:18) D + 2 R c R (cid:19) +2 R c (cid:20) RK − L (cid:21) ln (cid:18) R c R (cid:19) + ln (cid:18) L RK (cid:19)(cid:27) , (1)where K = 1 − R c /R + x (1 − ln x ), x ≡ ( D + 2 R c ) / R .Further, β = ( k B T ) − is the inverse temperature, N trap represents the number of spherically trapped ions, L isthe equilibrium brush height and R = R c + L . Theentropic force is given as F en ( D ) = − ∂V en ( D ) /∂D .The basic assumption underlying the derivation ofEq. (1) above is that of no interdigitation between thetwo brushes: as the surface-to-surface distance D be-comes smaller than 2 L , the chains of each brush retractto the half-space in which the respective colloidal corelies. The experimental data at hand, however, cannotbe described by the force derived from the entropic con-tribution of Eq. (1), because the resulting forces have acompletely different D -dependence than the experimen-tal ones (see, e.g., Fig. 4). This is a clear indicationthat a different physical mechanism is at play for thesystem at hand. The rather small grafting density of thebrushes brings about a different possibility, namely themutual interdigitation of the brushes up to a surface-to-surface separation L and the subsequent compression ofthe chains opposite to the hard colloidal core for smallerdistances. This mechanism has been clearly identifiedand quantitatively analyzed in Ref. [15], in which interac-tions of star-branched polyelectrolytes with hard, planarsurfaces have been discussed. Taking into account thatchains from both brushes get compressed against the coreof the opposite brush, the expression for the compres-sion contribution to the effective brush-brush interactionreads for d ≪ D ≤ L as βV c ( D ) = ( Z eff N ) λ B D × ( (cid:18) Dd (cid:19) + (cid:18) DL (cid:19) (cid:20) ln (cid:18) L d (cid:19) − (cid:21)) . (2)Here, the Bjerrum length λ B = βe /ǫ denotes the dis-tance at which the electrostatic energy equals the thermalenergy and has the value λ B = 7 .
18 ˚A for water at 300 K.Further, d is the typical diameter of individual arms ofthe two interacting brushes, having for DNA the value d = 18 ˚A [16, 17]. Again, the compression contributionto the force is given as F c ( D ) = − ∂V c ( D ) /∂D and it canbe easily checked that F c ( D ) vanishes at D = L . Con-trary to the entropic contribution, which sets in whenthe coronae overlap, i.e., at D = 2 L , the compressioncontribution requires that the grafted chains of one brushtouch the core colloid of the other and thus it is nonvan-ishing in the range D ≤ L .The brush height L used in the theoretical calcula-tion of the forces is read off from experiment and it isthus no free fit parameter. The ‘effective ion valency’ Z eff appearing in Eq. (2) is treated as a fit parameter tothe experimental data; nevertheless, it is constrained bycertain physical considerations based on known facts onthe propensity of DNA strands to adsorb and stronglycondense counterions on their grooves. Indeed, as it hasbeen shown experimentally [18, 19], DNA can stronglycondense about 90% of counterions, which already sets arough upper limit of 0 . Z eff to evenlower values. All parameter combinations for the various I [M]100 λ F = p N [ n m ] LaCl CaCl NaClOsmotic brush Salted brushSlope ~ -1/320020
FIG. 3: Double-logarithmic plot of the interaction lengthfor a force F = 2 pN versus the ionic strength of the addedsalt. Here, the molecular weight of the grafted DNA is N =1000. Different types of symbols correspond to different saltvalencies. The dashed line of slope − / D [nm]0246810 F [ p N ] σ =1.8·10 -4 nm -2 σ =5.9·10 -5 nm -2 σ =2.0·10 -5 nm -2 FIG. 4: Effective force-distance curves for DNA-grafted col-loids at different grafting densities σ , fixed molecular weight N = 1000 bp and no added salt. Symbols: experiment; lines:theory. For the lowest σ , the solid line also includes an en-tropic contribution to force, derived from Eq. (1), which isillustrated by the dashed line. DNA-grafted colloids are shown in Table I.The experimental forces shown in Figs. 1 and 2 arevery well described by the compression force; there arestill ‘tails’ for
D > L that can be discerned, and whichcorrespond to small contributions from the entropic force,Eq. (1). The purpose of this work lies in the understand-ing of the compression-induced forces, thus we have notattempted a detailed description of the tails, especiallysince the magnitudes of the resulting forces there lie atthe limit of experimental accuracy. We show neverthe-less for the lowest grafting density in Fig. 4 a typicalexample of the combination of compression and entropiccontributions that results into an excellent descriptionof the forces. Here, a number of N trap = 1100 spheri-cally trapped counterions was employed, which is in goodagreement with the simulation results of Ref. [20]. There,planar PE-brushes were simulated and for a grafting den-sity very close to the lowest one in Fig. 4, it was foundthat about 10% of the counterions are outside the brush. TABLE I: The physical parameters of the employed DNA-grafted colloids and the effective valency Z eff employed in thetheoretical modeling of each system.Figure σ [nm − ] Base pairs I [mM] L [nm] Z eff . · −
250 0.0 100 0.227 a
500 130 0.115750 180 0.1041000 280 0.091 b . · − c . · − . · − c . · − . · −
310 0.0562 . · −
180 0.011 a Except for this value for Z eff , which is still of the order 0.1, allother lie below the threshold 0.1 mentioned in the text. Note, how-ever, that here L deviates from linear scaling with N , as mentionedbefore, and this fact may affect the precise Z eff -value. b Here, different colloids were used than in the cases marked withthe superscript c below, which explains the deviation between thecorresponding experimental results and the concomitant differencein the Z eff -values. c In these two cases, the same colloids were used, yet the measuredforces show minimal differences set by experimental accuracy. Thetheoretical values of Z eff are then slightly different between thesetwo cases. As the vast majority of the remaining 90% is Manning-condensed on the rods, a very small relative number ofabout 1000 spherically condensed ones, results, in agree-ment with the value of N trap mentioned above. For thehigher grafting densities shown in Fig. 4, the entropiccontribution seems to be negligible.Summarizing the results shown in Figs. 1, 2, and 4, itcan be surmised that the compression force resulting fromEq. (2) yields a very good description of a large varietyof experimental data. The effective valency Z eff fromTable I always lies in the physically expected region andshows the expected dependence on salinity, decreasingwith ionic strength I . Note that already the fact thatthe resulting forces from theory lie at the pN-domain isa nontrivial feature, in view of the fact that quantities ofvastly different order of magnitude in SI (Bjerrum length,Boltzmann constant, brush height and DNA-diameter) are involved in determining its numerical value.We have measured and theoretically described theforces between spherical DNA-brushes with low graftingdensity. The physical system at hand provides a convinc-ing verification of the importance of the PE-compressionmechanism [15], in sharp contrast to most hitherto stud-ied systems, which were dominated by counterion en-tropy. Therefore, it has been demonstrated that thepresent systems are colloids whose effective interactionis short-range, i.e., tunable in terms of its extension andstrength by changing the number of base pairs involved,the ionic strength and grafting density. The quantitativecharacteristics of the resulting effective force are unique:whereas for neutral, densely grafted brushes the forcescales as F ( D ) ∼ D − for D ≪ L [21, 22], here a depen-dence F ( D ) ∼ D − ln( D/d ) for d ≪ D ≪ L results. Onthe other hand, for D < ∼ L , we obtain F ( D ) ∼ | D − L | .Future work should focus on the study of concentratedsolutions of such brushes, including crystal- and glass for-mation, and the analysis of these in terms of the effectiveinteraction derived in this work.We thank Dr. Arben Jusufi (Princeton) for helpful dis-cussions. This work has been supported by the DFG.C.N.L. wishes to thank the ESI (Vienna), where parts ofthis work have been carried out, for its hospitality. [1] P. Bolhuis and D. Frenkel, Phys. Rev. Lett. , 2211(1994).[2] E. A. Jagla, J. Chem. Phys. , 451 (1999).[3] Y. Norizoe and T. Kawakatsu, Europhys. Lett. , 583(2005).[4] M. A. Glaser et al. , Europhys. Lett. , 46004 (2007).[5] Z. Yan et al. , Phys. Rev. E , 051204 (2006).[6] P. Kumar et al. , Phys. Rev. E , 021501 (2005).[7] J. Largo, P. Tartaglia, and F. Sciortino, Phys. Rev. E ,011402 (2007).[8] P. Pincus, Macromolecules , 2177 (1991).[9] O. V. Borisov and E. B. Zhulina, Eur. Phys. J. , 205(1998).[10] Y. Mei et al. , Phys. Rev. Lett. , 158301 (2006).[11] A. Jusufi, C. N. Likos, and M. Ballauff, Colloid Polym.Sci. , 910 (2004).[12] A. Jusufi, C. N. Likos, and H. L¨owen, Phys. Rev. Lett. , 018301 (2002); J. Chem. Phys. , 11011 (2002).[13] K. Kegler, M. Salomo, and F. Kremer, Phys. Rev. Lett. , 058304 (2007).[14] Y. Rabin, G. H. Frederickson, and P. Pincus, Langmuir , 2428 (1991).[15] M. Konieczny and C. N. Likos, J. Chem. Phys. ,214904 (2006).[16] H. M. Harreis et al. , Phys. Rev. Lett. , 018303 (2002).[17] A. A. Kornyshev et al. , Rev. Mod. Phys. , 943 (2007).[18] V. A. Bloomfield, Curr. Opin. Struct. Biol. , 334 (1996).[19] A. A. Kornyshev and S. Leikin, Phys. Rev. Lett. , 4138(1999).[20] H. Fazli et al. , Europhys. Lett. , 429 (2006).[21] S. T. Milner, T. A. Witten, and M. E. Cates, Macro- molecules , 2610 (1988).[22] J. Mewis et al. , AIChE. J.35