The effective hydrodynamic radius of single DNA-grafted colloids as measured by fast Brownian motion analysis

Polymer 52 (2011) 1829e1836

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The effective hydrodynamic radius of single DNA-grafted colloids as measured by fast Brownian motion analysis Olaf Ueberschär*, Carolin Wagner, Tim Stangner, Christof Gutsche, Friedrich Kremer Institut für Experimentelle Physik I, Universität Leipzig, Linnéstraße 5, 04103 Leipzig, Germany

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 November 2010 Received in revised form 31 January 2011 Accepted 1 February 2011

Optical tweezers accomplished with fast position detection enable one to carry out Brownian motion analysis of single DNA-grafted (grafting density: w1000 molecules per particle, molecular weight: 4000 bp) colloids in media of varying NaCl concentration. By that the effective hydrodynamic radius of the colloid under study is determined and found to be strongly dependent on the conformation of the grafted DNA chains. Our results compare well both with recent measurements of the pair interaction potential between DNA-grafted colloids (Kegler et al. Phys Rev Lett 2008; 100:118302) and with microfluidic studies (Gutsche et al. Microfluid Nanofluid 2006; 2:381-386). The observed scaling of the brush height with the ion concentration is in full accord with the theoretical predictions by Pincus, Zhulina, Birshtein and Borisov.
2011 Elsevier Ltd. All rights reserved.

Keywords: Colloidal polymer brush Polyelectrolyte brush height scaling Brownian motion analysis

1. Introduction Polymer layers that are grafted to microsphere surfaces, socalled colloidal polymer brushes, are of great interest in several fields of current material science. The radius of the used beads usually ranges from 0.5 mm to 2 mm. Colloidal polymer brushes show intriguing macroscopic properties that emerge from their unique microscopic surface structure and the characteristics of its interaction with the surrounding fluid. The scientific and technological application of these properties comprises, for instance, the improvement of colloidal stability [1], the preparation of biodegradable lubricants [2] as well as basic research on microscopic and molecular friction. Detailed studies of the physical properties of such polymergrafted surfaces have been carried out by means of experimental [3e19], analytical [20e27] and numerical approaches [2,28e30]. The previously employed experimental methods include dynamic light scattering, small-angle neutron and X-ray scattering as well as cryogenic transmission electron microscopy [4]. In essence, all these studies show that the interaction of the polymer layer with the surrounding fluid, especially under shear flow, plays a key role in the understanding of the physical properties of the surface. The influence of the shear flow on polymer brushes has been thoroughly

* Corresponding author. Permanent address. Institut für Experimentelle Physik I, Fakultät für Physik und Geowissenschaften, Universität Leipzig, Linnéstraße 5, 04103 Leipzig, Germany. Tel.: þ49 341 9732718; fax: þ49 341 9732599. E-mail address: [email protected] (O. Ueberschär). 0032-3861/$ e see front matter
2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2011.02.001

discussed for steady-state conditions [11,32]. It has been revealed therein that the polymer brushes significantly attenuate the surrounding shear flow by viscous drag. The fluid flow is affected in such a way that it stagnates at a certain finite height, the so-called stagnation height, above the surface. These “lift up” and “no slip” properties of the polymer layer are of eminent relevance for the (micro-)rheology of the grafted surface. In analogy, polymer-grafted colloids show a hydrodynamic radius that is significantly increased with respect to the blank colloid [32], both under shear flow and thermal equilibrium conditions. In recent experimental studies, polyelectrolytes are often used for surface grafting [19,31e35]. Polyelectrolytes, such as DNA molecules, are polymer macromolecules whose monomers carry ionizable functional groups, which dissociate in polar solvents, such as water. This property is characterized by the degree of ionization a. If the latter is independent of the pH value, the polymers are referred to as quenched polyelectrolytes. If, however, the degree of ionization depends on the pH value of the surrounding fluid, the polymers are called annealed polyelectrolytes [7,9,10,36]. On the basis of these properties, polyelectrolytes that are grafted, for instance, to colloids show different spatial conformations. These conformational states are classified as “pancake”, “mushroom” and “brush” regimes. Depending on the grafting density s (number of macromolecules per area unit of the surface) and the degree of ionization a, always one of these three conformations is adopted [37]. Theoretical treatises of polymer brushes are commonly based on the work by Pincus [20], Zhulina, Borisov and Birshtein [21,23,24]. They found scaling laws for the height of polyelectrolyte brushes with respect to different salt concentrations of the

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O. Ueberschär et al. / Polymer 52 (2011) 1829e1836

surrounding medium. In particular, there exist an osmotic regime and a salted regime. In the osmotic regime of a planar polyelectrolyte brush, the brush height generally H scales as

HyNaa1=2

(1)

where N denotes the number of monomer units per polymer chain and a the size of such a monomer. In the salted regime, however, H scales as

1=3
HyN a2 a2 s1 c1

(2)

where c is the salt concentration [23]. The DNA macromolecules that constitute our brush layer represent quenched polyelectrolytes [38]. For them, the degree of ionization a is fixed. For annealed polyelectrolytes, however, the situation is generally more complicated. As a depends on the salt concentration in the annealed case, different scaling laws are obtained in dependence of the ionic strength: One can show that for low ionic strengths, the brush height scales with the salt concentration as Hfcþ1=3 [23]. For high ionic strengths, in contrast, the degree of ionization a becomes practically constant as for quenched polyelectrolytes. As a result, the difference between the annealed and quenched brushes disappears for high salt concentrations, yielding a common scaling as Hfc1=3 [23]. As regards geometry, the scaling exponent g ¼ 1=3, i.e. Hfcg , is rigorously valid only for planar surfaces. If the geometry is changed to a star-like polymer brush, an analogous scaling law can be derived, now yielding an exponent of g ¼ 1=5 [25]. Both planar and star-like brushes may be interpreted as limiting cases of a colloidal polymer brush consisting of a solid core with radius R0 and a polymer layer consisting of grafted polyelectrolyte chains with the contour length L. The exact value of the exponent g then depends on the ratio R0 =L : g ¼ 1=5 if R0  L , g ¼ 1=3 if R0 [L and g˛1=5; 1=3½ otherwise [23,25,26]. In spite of these detailed findings, theory still lacks a general comprehensive description of the hydrodynamic properties of polymer-grafted surfaces. By now, the cited scaling laws for the osmotic and the salted regime could be investigated experimentally on a single particle level by measuring pair interaction forces between two DNA-grafted colloids [32,33] and by shear flow experiments with a single DNA-grafted microsphere [31]. The results of the latter experiments were cross-checked and confirmed by hydrodynamics simulations on the basis of the semiflexible chain model [11]. For mean value studies, dynamic light scattering experiments carried out in solutions with many similar particles revealed interesting insights [4]. In the following sections, we will present and discuss a further approach for measuring the hydrodynamic radius of a single DNAgrafted polystyrene microsphere. The proposed method is based on the two-dimensional optical tracking of a singe colloid in different media, i.e., different salt concentrations, while it is being held in an optical trap. The analysis of the Brownian motion of the trapped colloid in thermal equilibrium yields precise information about the effective hydrodynamic radius. 2. Methods and materials 2.1. Radius measurement method In order to give a brief recapitulation of the applied theoretical concepts, we consider a microsphere dissolved in a liquid and held P in the harmonic potential VðxÞ ¼ 12 3i ¼ 1 ki x2i of an optical trap. The system is in thermal equilibrium, so that the Brownian motion of the microsphere (mass m, Radius R, drag coefficient x) obeys the Langevin equation

xx_ i þ ki xi ¼ FiðsÞ ðtÞ; ci˛f1; 2; 3g

(3)

where the index i˛f1; 2; 3g denotes the respective space dimension [44,45]. As usual, the inertia term is neglected as the corresponding time scale is much shorter than the experimentally accessible ðsÞ temporal regime. Fi ðtÞ is the time-dependent stochastic force. For an ideal, non-rotating spherical particle of radius R in an incompressible fluid of infinite volume and with a temperature-dependent dynamic viscosity hðwÞ , the drag coefficient x is given by the Stokes formula, i.e. x ¼ xðwÞ ¼ 6phðwÞR. A polystyrene colloid dissolved in a water-filled fluidic cell fulfills these assumptions as long as the distance h of the microsphere to all cell boundaries is much greater than its radius, i.e. h[R. Otherwise, the Faxén correction term must be introduced1 in x ¼ xðh; RÞ [47,48]. The temperature dependence of the dynamic viscosity hðwÞ of pure water under standard conditions is empirically given by [40,49]




~ h w lg

¼

Pa s



2 ~  20  0:001053  w ~  20 1:3272  w ~ þ 105 w  2:999

(4)

C ~ whw=

where and lgxhlog10 x. The (non-normalized) autocorrelation function of the coordinate xi can be derived from equation (3), yielding [45,46]

1 t0 /N t0

Cxi ;xi ðtÞ : ¼ hxi ðt 0 Þxi ðt 0 þ tÞi ¼ lim

Zt0

D E k T ¼ B ejtj=si ¼ x2i ðt 0 Þ ejtj=si ki

xi ðt 0 Þxi ðt 0 þ tÞdt 0

0

ð5Þ

Normalization is accomplished by dividing through by the variance hxi i ¼ kB T=ki , as provided by the equipartition theorem. Here, T denotes the absolute temperature, i.e. w= C ¼ T=K  273:15. The characteristic time constant si may be expressed in terms of the drag coefficient a and the trap stiffness ki :

si : ¼

x ki

¼

6phðwÞR ki

(6)

For typical values of the parameters temperature, viscosity, bead radius, and trap stiffness, e.g. w ¼ 20 C , hð20 CÞ ¼ 1002 mPa s , R ¼ 1:1 mm , ki ¼ 4 fN nm1 , a time constant of si z4 ms is obtained. In combination with the relation si fki , it is preferable to carry out measurements of this autocorrelation time constant si with the lowest reasonable trap stiffness available in order to achieve optimum signal quality. As revealed by Eq. (6), si depends on both the radius R and the temperature w. This expression is used in the following for determining the radius of a trapped colloid. The temperature is measured by means of a thermocouple near the focus position. Heating effects within or near the colloid have been ruled out by a separate consistency check. For this check, Eq. (6) was solved for the temperature w during a continuous external cooling process while the radius R was known from a separate measurement carried out before. For the radius measurement process, the colloid is held in the optical trap with a stiffness of ki w4 fN nm1 (corresponding to a laser power in front of the objective of about 10 mW in our setup). The Brownian motion of the microsphere is tracked for 2 s with a frame rate of 5 kHz, yielding a two-dimensional trajectory data array of 1  104 single coordinate pairs. These trajectory data are

1 However, for h > 40R this correction has a contribution of less than 1.5% and is therefore then negligible.

O. Ueberschär et al. / Polymer 52 (2011) 1829e1836

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Fig. 1. (a) Sketch of the utilized experimental setup. (b) The sample cell as described in the text. (c) Schematic of a streptavidin-coated microsphere to which DNA molecules are grafted.

analyzed dimension-wise for their variance hxi ðt 0 Þi and their autocorrelation function Cxi ;xi ðtÞhhxi ðt 0 Þxi ðt 0 þ tÞi, where the angle brackets hi are evaluated over running discrete starting time indexes t 0 as permitted by ergodicity. In order to allow for an undesirable thermal drift of the laser focus, the centre of the optical trap is re-defined by forcing the mean hxi ðt 0 Þi to vanish. The significant first 2.5 ms of the autocorrelation function Cxi ;xi ðtÞ, comprising approximately 12e25 data points, are fitted to the theoretically predicted exponential form of Eq. (5). In a semi-logarithmic plot (logarithmic ordinate), the functional relation is represented by a straight line as depicted in Fig. 3. Its slope equals si whereas the offset is equal to lnhxi i. Thus, a linear fit yields the two sought quantities si and hxi i. With the temperature w being measured by means of the thermocouple, the radius R of the trapped colloid is simply obtained by rearranging Eq. (6), reading

Ri ¼

si ki 6phðwÞ

(7)

where the dynamic viscosityhðwÞ of water2 is obtained from Eq. (4) and the spring constant ki is deduced from the variance hx2i ðt 0 Þi by applying the equipartition theorem. Altogether, about 20e30 trajectories are recorded and analyzed in the presented fashion in order for the confidence level to settle at an acceptable value (cf. Fig. 3). The measurement uncertainty of the presented method is derived and discussed in the appendix. There, it is carefully evaluated to 1:2%. 2.2. DNA grafting Streptavidin (SA)-modified polystyrene beads with a diameter of 2.18 mm were purchased from Polysciences Europe (Eppelheim, Germany). The doublestranded 4000-bp long DNA is amplified from the plasmid pET28a þ via polymerase chain reaction (PCR) and labeled with digoxigenin (DIG) or biotin, respectively, at the termini via 50 -end modifications of the primers (Metabion, Matinsried,

2 Dispersing agents other than water can be allowed for by relations for hðwÞ analogous to Eq. (4).

Germany) used for PCR: (50 -Dig-CAG CTT CCT TTC GGG CTT TG-30 ) and (50 -Bio-TGA TTG CCC GAC ATT ATC GC-30 ). Nucleic acid purification kits were supplied by QIAGEN (Hilden, Germany). All enzymes and standards were obtained from MBI Fermentas (Vilnius, Lithuania). All other chemicals were delivered from Sigma (Deisenhofen, Germany). The grafting of the SA-covered particles with functionalized DNA (cf. Fig. 1, c) is done by the following protocol: 10 ml of SA microspheres are centrifuged with 10 000 g. The resulting supernatant is discarded. In the next step, the particles are re-suspended in 100 ml PBS buffer (pH 7.4) and centrifuged again. Subsequently, the beads are re-suspended in 10 ml PBS buffer and an amount of 7.6 ml of DNA-solution is added. The whole mixture is incubated for 30 min at room temperature. Afterward, the solution is centrifuged again with 10 000 g, the supernatant is removed and its DNA concentration is measured by employing a NanoDrop 2000 analyzer (Thermoscientific NanoDrop products, Wilmington, USA). For providing a reference value, 10 ml of PBS buffer are added to further 7.6 ml of the same DNA-solution as before. The resulting DNA concentration is determined analogously by means of the NanoDrop. From the difference of those two DNA concentrations the amount of DNA bound to the surface of the microparticles can be determined. Under the assumption that in all microspheres have bound approximately the same amount of DNA, a mean grafting density of w1000 DNA molecules per particle is obtained. For the last preparation step, the DNA-grafted beads are resuspended in 100 ml Tris buffer and washed twice with 100 ml Tris each. Then, the particles are re-suspended in 10 ml Tris buffer. This final suspension is incubated overnight at 4  C. Further details about the presented protocol can be found in [51]. 2.3. Experimental setup The utilized optical tweezers setup (Fig. 1 a) is based on an inverted microscope (Axiovert 200, Carl Zeiss Jena, Germany) that is equipped with a water-immersion objective (Olympus UPlanApo/IR, 60 x, NA ¼ 1:20) and an external fiber cold light source. A piezo stage (P-562.3, Physik Instrumente, Karlsruhe, Germany) being controllable in all three space dimensions with nanometre-accuracy is installed. The optical trap is formed by a highly focussed Nd:YAG

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O. Ueberschär et al. / Polymer 52 (2011) 1829e1836

Fig. 2. (a) Contrast-enhanced false-colour diffraction image of a R ¼ 1.5 mm colloid as it is taken by the CMOS camera with a subsequent optical magnification of 2.5 and a frame rate of 5 kHz. (b) The corresponding intensity matrix is plotted as a three-dimensional surface, which is fitted to the model function of Eq. (8) (depicted as a red wire frame). The z-axis is scaled in absolute digital intensity units ranging from 0 to 255 according to an image bit depth of 8. (c) The centre of the colloid as determined by the described fit is subject to Brownian motion, which becomes apparent from the depicted trajectory.

laser beam (TEM00 , l0 ¼ 1064 nm , LCS-DTL 322, Laser 2000, Wessling, Germany). The laser power can be adjusted in the range of 5 mW to 1000 mW. A PID closed-loop control circuit ensures temporal stability during measurements. The force calibration of the trap is accomplished by applying the equipartition theorem. A water-filled clamped sample cell (Fig. 1 b) with an inner volume of 0.1 ml is mounted on the piezo stage. Because the immersion water is subject to continuous evaporation during measurements, a remote-controlled water pump is installed to keep the immersion liquid volume at a sufficient level. The two-dimensional particle tracking is based on digital greyscale CMOS video camera imaging with a maximum frame rate of 10.1 kHz (MC1310, Mikrotron GmbH, Unterschleissheim, Germany) in combination with a diffraction pattern fitting algorithm. As in other pertinent work [53], this analysis algorithm fits the empirical, rotationally symmetric model function 0

0




3

0



0

Iðr Þ ¼ I0 þ A$ð1  auðr ; rÞÞ exp  u ðr ; rÞ ; uðr ; rÞ : ¼

jr0  rj2 0 T ; r ¼ ðx0 ; y0 Þ R0

(8)

to the measured planar intensity profile by an implementation of the LevenbergeMarquardt algorithm [50]. The quantities I0 , A, a and R0 are free fit parameters (disregarded in the following) whereas r ¼ ðx; yÞT represents the retrieved colloid’s centre coordinates. By this image analysis, changes in the trapped colloids xand y-coordinates (see Fig. 2) are detected with an accuracy of 4 nm. Because the measured planar intensity profile has been verified to be an orthogonal projection image of the colloid, the (disregarded) z-axis displacement does not affect the planar noise analysis in x- and y-direction.

colloid was trapped with a trap stiffness of ki w4 fN nm1 . Its radius was determined as discussed above, yielding a value of ð1:085  0:013Þmm. This value lies within the range of the manufacturer’s radius specification of the batch, i.e. ð1:09  0:01Þmm. Then, we flushed the sample cell with a 10 mM Tris buffer solution and afterward with a 10 mM Tris solution containing 1 M NaCl while the trapped colloid was being held stably in the trap with an increased stiffness of ki w100 fN nm1 . After either flushing process, we measured the autocorrelation constant si . As in all three media (millipore water, 10 mM Tris, 10 mM Tris with 1 M NaCl) the same colloid was measured at the same system temperature of 20.0  C, the obtained differences in si are entirely due to the changed viscosity. The viscosity values determined in this way are in good agreement with the literature data, see Table 1. 3.2. Total increase of the effective hydrodynamic radius of a single colloid by DNA grafting Second, we carried out an experiment with a single DNA-grafted colloid. The sample cell was filled with a 10 mM Tris buffer solution and a few DNA-grafted colloids were flushed in. One of these microspheres was trapped with a trap stiffness of ki w4 fN nm1 . While it was being held in the optical trap, we determined its radius as before. The viscosity of 10 mM Tris being virtually the same as for water (cf. Table 1), we obtained a value of Rgrafted ¼ ð1:217  0:011Þmm. Afterward, the sample cell was flushed with a 10 mM Tris solution containing 1 M NaCl at ki w100 fN nm1 . The addition of monovalent NaCl in a concentration of 1 M effects a total collapse of the DNA brushes [32,33]. The different viscosity of this salty Tris buffer solution taken into account (cf. Table 1), a radius of RDNA collapsed ¼ ð1:082  0:012Þmm was measured for the state of

3. Results and discussion 3.1. Viscosimetry by means of blank colloid In order to confirm the literature values of the dynamic viscosity of the buffers to be used for DNA-grafted colloids, we first employed the presented Brownian motion analysis method in combination with a blank colloid for viscosimetry. A blank streptavidin-coated

Table 1 Viscosity values of the used fluids for w ¼ 20:0 C. Fluid

Literature value of hðwÞ/(mP s)

Measured value of hðwÞ/(mPa s)

Millipore water Tris 10 mM Tris 10 mM with 1 M NaCl

1002 1002 1104

e 1002  10 1110  10

O. Ueberschär et al. / Polymer 52 (2011) 1829e1836 80

60 Cxx (Tris 10 mM)

1400

Cyy (Tris 10 mM)

Cxx (Tris 10 mM with 1 M NaCl)

1300 1200 0.0

Cyy (Tris 10 mM with 1 M NaCl)

0.2

0.4

0.6

0.8

1.0

1.2 1.4 t / ms

b

1.8

2.0

2.2

2.4

y(t ) / nm

y (t ) / nm

-100

2

S(f ) / (nm s)

S(f ) / (nm s)

10

100 f / Hz

1000

30

0 100 x (t ) / nm

3

10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1

10

100 f / Hz

1000

mean-square distance of the monomers [42,43]. In this respect it is noteworthy that the remaining thickness of the totally collapsed polymer layer is finite and is the range of a few nanometres [32,33]. Fig. 3 shows the autocorrelation functions, examples of measured trajectories and the power spectra of the used colloid in the states of the initial “osmotic brush” regime (10 mM Tris) and the entirely collapsed brush regime (10 mM Tris with 1 M NaCl). 3.3. Average total increase of the effective hydrodynamic radius by DNA grafting Furthermore, the distribution of radii of blank and DNA-grafted colloids was investigated for a sample size of 40 colloids each. The used buffer solution was again Tris 10 mM. For this measurement, however, no flushing was necessary. Instead 40 blank colloids and

120

is about 1.3 times the radius of gyration Rg of the used doublestranded DNA molecules (4000 bp, base pair length of 0.34 nm, persistence length of z50 nm), reading [41]

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð4000$0:34 nm$50 nmÞz106 nm 6

(10)

If this hydrodynamically effective polyelectrolyte layer thickness DR is to be interpreted to first approximation as the mean-square distance of the DNA chain units frompthe ffiffiffiffiffiffiffiffisolid colloid surface, it should be in the range of Rg  DR  5=3Rg z1:3Rg according to the theory of Birshtein and Borisov [21,22]. Obviously, this interpretation is qualitatively reasonable. However, the hydrodynamically effective layer thickness is in general smaller3 than the

Δ R / nm

(9)

c / (mol l )

100

100

DR ¼ Rgrafted  Rcollapsed ¼ ð135  22Þnmz1:3Rg

-1

140

totally collapsed DNA. This value corresponds to the mean radius of the used blank strepatividin-coated colloids, i.e. ð1:09  0:01Þ mm. Thus, the decrease DR of the radius due to the addition of 1 M NaCl,

For a sphere the factor is 3p1=2 =8z2=3.

1.00

Fig. 4. Histograms of the distribution of radii of blank streptavidin-coated colloids (blue) and DNA-grafted colloids (red). For each species, 40 colloids were measured by means of the presented Brownian motion analysis method. Obviously, the manufacturer’s specification (green line with surrounding box) of the mean radius is confirmed by our measurement. For reasons of clarity, the displayed error labels in the graph represent the standard deviations of the respective distribution, not the standard error of the mean as in the text. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

Fig. 3. (a) Example of a position autocorrelation function Cxi xi ðtÞ obtained from one trajectory each for the same DNA-grafted colloid in Tris 10 mM (blue) and Tris 10 mM with 1 M NaCl (green). The measurement parameters are N ¼ 104, frame rate f ¼ 5.1 kHz, ki z4 fN nm1 and w ¼ 23:5 C. The 1 M NaCl solution effects a total collapse of the DNA brushes. The viscosity of this solution is 10.8% higher than that of the pure Tris 10 mM buffer (cf. Table 1). The noted radii are the mean results of 30 trajectories, as described in the text. (b) and (c) One example of one trajectory each from the data of graph (a). (d) and (e) Mean power spectra of the 30 trajectories each. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

Rg ¼

0.95

0

-100

e

1

40

R / µm

0 100 x (t ) / nm

3

10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10

Mean: (1.203 ± 0.054) µm

113 nm

50

0 0.90

-100 -100

Mean: (1.090 ± 0.026) µm

10

100

0

Blank DNA-grafted (4000 bp)

20

c 100

2

1.6

R = (1.217 ± 0,011) µm R = (1.082 ± 0.012) µm

Tris 10 mM: Tris 10 mM with 1M NaCl:

d

Manufacturer: (1.090±0.010) µm

70

(1) (2)

80

Slope = -0.27 ± 0.01 Δ R / nm

1800 1700 1600 1500

Rel. frequency / %

Cxx(t), Cyy(t)

a

3

1833

10

60

-4

40

-3

10

-2

10

10

20 0 0.0

-3

1.0x10

-2

10

-1

10

0

10

-1

c / (mol l ) Fig. 5. Increase of the hydrodynamic radius with respect to different NaCl concentrations as measured for a single DNA-grafted colloid. The molecular weight of the DNA molecules is 4000 bp. The inset shows the data points (to logarithmic scale) of the salted regime, which is present here for concentrations of 30 mM, 100 mM, 300 mM, 1 mM and 3 mM. The linear fit yields a scaling exponent of g ¼ ð0:27  0:01Þ. Reversibility was checked both after 100 mM (blue triangular data point with label “(1)”) and after 1 M (“(2)”) as described in the text.

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O. Ueberschär et al. / Polymer 52 (2011) 1829e1836

40 DNA-grafted colloids were measured consecutively for each species. The resulting histograms are presented in Fig. 4. For the blank streptavidin-coated colloids, we obtained a mean radius of ð1:090  0:004Þmm and a standard deviation (SD) of the distribution of 0.026 mm. Hence, this measurement exactly reproduces the manufacturer’s mean value of 1.090 mm while it yields a roughly doubled SD. Possible reasons for the latter are discussed in Section 6 below. For the DNA-grafted colloids, the measurement yielded a mean radius of ð1:203  0:009Þmm and a SD of 0.054 mm. Thus, the average increase hDRi of the effective hydrodynamic radius due to DNA-grafting is evaluated to

Relative frequency / %

20

15

Manufacturer: (m) = 1.09 µm (m) = 0.01 µm σ

Samples from batch no. 1

Measurement: = 1.093 µm = 0.018 µm <σ >

10

5

    hDRi ¼ Rgrafted  Rblank ¼ ð113  13ÞnmzDR 0 0.95

1.00

1.05

1.10

1.15

1.20

(11)

this agrees with the value DR obtained in the single colloid experiment (cf. Eq. (9)) within measurement uncertainty. Moreover, the measured SD of the DNA-grafted colloids is about twice the SD of the blank colloids. This broadening may be understood in the context of preparation variability: Although valid in average, it is unlikely that each single colloid has bound exactly the same number of DNA molecules on its surface during the grafting process (cf. Section 2.2). Moreover, the geometry of the resulting DNA osmotic brushes on the surfaces of the colloids is subject to certain statistic variability.

1.25

R / µm Fig. 6. Comparison of a measured sample to the manufacturer’s specifications: 50 single colloids from one batch (referred to as no. 1 in Table 2) were trapped and measured by means of the presented method. The resulting histogram roughly shows a Gaussian distribution around a mean value of hRi ¼ 1:093 mm and a SD of s ¼ 0.018 mm. Both values are in quantitative agreement with the corresponding manufacturer’s value.

3.4. Dependence of the brush height on the salt concentration Table 2 Comparison of three different measured average radii to manufacturer’s specifications.

1 2 3

Manufacturer

Measurement

hRiðmÞ =mm

sðmÞ =mm

hRi=mm

s=mm

1.09 z1.1 1.50

0.01 n/a 0.05

1.09 1.17 1.52

0.02 0.04 0.04

x / nm

a

b

200 150 100 50 0 -50 -100 -150 -200

Time series Rel. frequency / %

Batch no.

In extension of the single colloid experiment described in (b), a single DNA-grafted colloid was consecutively measured in nine different 10 mM Tris buffer solutions with a NaCl concentration of 0 M, 10 mM, 100 mM, 300 mM, 0 M, 1 mM, 3 mM, 1 M and 0 M. The consequent use of the same colloid ensures identical material parameters (such as colloidal core radius and polymer brush grafting density) throughout the experiment. The runs in pure Tris

0,0

0,5

1,0

1,5

8 7 6 5 4 3 2 1 0

Histogram

Δ x = 6πη Rv / kx

2,0

-50

t/s

R / µm

c

2,0 1,9 1,8 1,7 1,6 1,5 1,4 1,3 1,2 1,1 1,0

0

50

100

150

x / nm

Viscous drag vs. noise measurement

Noise analysis with confidence level Viscous drag measurement with error bars

0

5

10

15

20

25

30

35

40

45

-1

v / µm s

Fig. 7. Comparison of a single blank colloid radius measurement by means of viscous drag and thermal noise analysis. (c) The blue line depicts (along with its surrounding confidence level) the radius value measured by means of the presented noise analysis method whereas the red spheres show the viscous drag experiments results for the same microsphere. The red error bars reflect the statistical error for 36 independent measurements per speed value. The upper two graphs (a) and (b) illustrate how such a single measurement point is obtained. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

O. Ueberschär et al. / Polymer 52 (2011) 1829e1836

buffer, i.e. with a concentration of 0 M of NaCl, that were carried out after the 300 mM NaCl and the 1 M NaCl run serve as checks for the reversibility of the conformational changes observed (see below). In each medium, the Brownian motion analysis was carried out with ki w6 fN nm1 while the media were exchanged again at ki w100 fN nm1 . The measured increase in the hydrodynamic radius with respect to the mean radius of a blank colloid is plotted versus the respective salt concentration in Fig. 5. The change of the viscosity of the 10 mM Tris buffer solution due to salt addition is accounted for by the empiric formula [52]

hðcÞ ¼ ð1:002 þ 0:09496$ðc=MÞÞmPa s

(12)

At a salt concentration of 3 mM, the measured increase of the hydrodynamic radius with respect to the blank colloid is 9 nm and therefore already at the detection limit of our radius measurement method. Consequently, measurements in higher salt concentrations will not yield further quantitative insights. Nonetheless, the total collapse of the DNA brush at a salt concentration of 1 M is confirmed also in this experiment. The salted regime exists in the brush under study for NaCl concentrations of approximately c > 10 mM [33]. In this regime, i.e. c ¼ 30 mM to c ¼ 3 mM in Fig. 5, the brush height is found to obey the scaling law DRfcg with a decay exponent of g ¼ 0:27  0:01. This result agrees well with previous experimental work [33,54] and the theoretical prediction of g˛1=5; 1=3½ (see above). Interestingly, the ratio of the blank colloid radius of R ¼ 1090 nm to the DNA contour length of L ¼ 1360 nm is R=Lz0:8w1 , so that the intermediate scaling regime at RzL is investigated by the presented measurement [23]. It is noteworthy in this context that the value of g ¼ 0:27  0:01 is approximately the mean of 1/5 and 1/3. In order to check the reversibility of the conformational changes of the polymer brush layer, a salt-free pure Tris buffer solution was flushed in both after the c ¼ 100 mM and the c ¼ 1 M measurement. While after 100 mM full reversibility within measurement uncertainty can be observed, the 1 M NaCl solution obviously provoked a partially irreversible brush height reduction: The layer thickness after re-flushing with pure Tris buffer is found to be 34 nm lower than the initial brush height, see Fig. 5. 4. Conclusion A new experimental approach is described that enables one to determine the effective hydrodynamic radius of single polymergrafted colloids held in an optical trap of low stiffness. By means of fast particle tracking, the Brownian fluctuations of the colloid under study are analyzed. On the basis of the Langevin equation and Stokes’ law, its effective hydrodynamic radius is determined. By that the conformation of the grafted polymers can be investigated in dependence on the ionic strength of the surrounding medium. The method-demonstrated here for DNA-grafted colloids-is widely applicable for colloidal polymer brushes and offers a novel tool on a single particle level. Acknowledgments Financial support from the Graduate School BuildMoNa and the DFG priority programme SPP1164 is gratefully acknowledged. Moreover, OU wishes to thank Jörg Reinmuth and Viktor Skokow for technical assistance as well as two anonymous reviewers for valuable comments on the first manuscript. Appendix. Discussion of measurement uncertainty In order to provide a reliable analysis of the numerical accuracy of the employed radius measurement method, its systematic and random errors are investigated in detail in the following. As regards

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systematic errors, the autocorrelation function itself bears an uncertainty because it is affected in an additive fashion by any sort of noise (as, e.g., instrument noise and particle tracking routine noise). Therefore, the measured variance of the particle coordinate xi is composed of a “thermal” part hx2i ijth ¼ 12kB T related to Brownian motion and a “measurement noise” part hx2i ijmeas , i.e. hx2i i ¼ hx2i ijth þ hx2i ijmeas hx2i ijth . The systematic deviation hxi ijmeas in our setup was found to be dominated by the limited spatial resolution of the particle tracking routine. The resulting relative error in the autocorrelation function was determined to be 0.2% for the used time domain of the first 2.5 ms. The according uncertainty in si is less than 0.05%. In combination with a maximum error of 0.05 ms related to camera timing, the maximum uncertainty in si is less than 0.3%. By means of these values, the maximum systematic error in Ri can be given by a simple propagation of uncertainty analysis for Eq. (7):

         DRi  Ds Dki  h0 ðwÞwDw   þ þ    R   s   k   h  w  i i < 0:3% þ 0:5% þ 1:4$0:25%z1:2%

(13)

For a radius of R ¼ 1:1 mm , this corresponds to an absolute systematic uncertainty of 11 nm. In addition to this analytical evaluation of the accuracy, we have carefully verified the accuracy of the employed method in several different ways for non-coated PS microspheres: i. Statistical measurement noise for identical colloid: For 30 independent trajectories with a length of 1  104 coordinate pairs, each measured for the same colloid, the statistical error (expressed in terms of the SD of the frequency distribution of the resulting 30 radius values) was found to numerically correspond to the values of the systematic uncertainties in radius as described above (Fig. 3). ii. Cross-check by comparison of the results obtained for x and y dimension: Owing to the spherical shape of the colloid, the results for radius (R1 and R2) must be equal within experimental uncertainty. If they are not, however, their discrepancy bears clear evidence for a defective measurement process (due to dirt particles or local liquid flow within the measurement cell, for instance). iii. Comparison of the radius distribution of a certain sample size to the manufacturer’s specifications: It is instructive to compare the mean hRi and the SD s of a certain sample size to the manufacturer’s specifications (as already done for the blank streptavidin-coated colloids in Section 3.3). For this purpose, random samples of 50 microspheres each from three batches with various radii have been checked. The analysis shows a clear-cut agreement to the manufacturer-delivered quantities, which are based on light diffraction measurements (see Fig. 6 and Table 2). iv. Comparison to viscous drag experiment results: This experimental procedure is similar to the established drag-based trap calibration approach [55,56]: The displacement hxi i resulting form a plug flow around the microsphere caused by ðpiezoÞ is measured. a piezo stage movement with speed vi According to Stokes’s law, this displacement is given by ðStokesÞ ðpiezoÞ i ¼ 6phðwÞRvi ¼ ki hxi i in the stationary state, hFi which is reached after approximately 5si w20 ms [39]. Provided that hxi i is measured with a sufficient accuracy despite the colloid’s Brownian motion, the radius R can be calculated by means of this relation. The obtained values agree well with the results of the noise-based measurements, as depicted in Fig. 7. Interestingly, this plug flow experiment yields non-trivial results if extended to DNA-grafted colloids, which is thoroughly discussed in [31].

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References [1] Napper DH. Polymeric stabilisation of colloidal dispersions. London: Academic; 1983. [2] Lai PY, Binder PY. J Chem Phys 1993;98:2366. [3] Perahia D, Klein J, Wartburg S. Nature 1991;352:143. [4] Ballauff M. Prog Polym Sci 2007;32:1135e51. [5] Ballauff M, Borisov OV. Curr Opin Colloid Interface Sci 2006;11:316e23. [6] Schärtl W, Lindenblatt G, Strack A, Dziezok P, Schmidt M. Prog Colloid Polym Sci 1998;110:285e90. [7] Guo X, Ballauff M. Langmuir 2000;16(23):8719e26. [8] Mei Y, Lauterbach K, Hoffmann M, Borisov OV, Ballauff M, Jusufi A. Phys Rev Lett 2006;97:158301. [9] Currie EPK, Sieval AB, Fleer GJ, Cohen Stuart MA. Langmuir 2000;16 (22):8324e33. [10] Raphael E, Joanny JF. Europhys Lett 1990;13(7):623e8. [11] Kim YW, Lobaskin V, Gutsche C, Kremer F, Pincus P, Netz RR. Macromolecules 2009;42(10):3650e5. [12] Klein J, Kumacheva E, Mahalu D, Perahia D, Fetters LJ. Nature 1994;370:634e6. [13] Baker SM. Macromolecules 2000;33(4):1120e2. [14] Tirrell M, Kokkoli E, Biesalski M. Surf Sci 2002;500:61e83. [15] Butt HJ, Cappella B, Kappl M. Surf Sci Rep 2005;59(1e6):1e152. [16] Li C, Benicewicz BC. Macromolecules 2005;38(14):5929e36. [17] Groenewegen W, Lapp A, Egelhaaf SU, van der Maarel JRC. Macromolecules 2000;33(11):4080e6. [18] Korobko AV, Jesse W, Egelhaaf SU, Lapp A, van der Maarel JRC. Phys Rev Lett 2004;93:177801. [19] Dukes D, Li Y, Lewis S, Benicewicz BC, Schadler L, Kumar S. Macromolecules 2010;43:1564e70. [20] Pincus P. Macromolecules 1991;24(10):2912e9. [21] Birshtein TM, Borisov OV. Polymer 1991;35(5):916e22. [22] Birshtein TM, Borisov OV. Polymer 1991;35(5):923e9. [23] Zhulina EB, Birshtein TM, Borisov OV. Macromolecules 1995;28(5): 1491e9. [24] Rabin Y, Alexander S. Europhys Lett 1990;13(49). [25] Klein Wolterink J, Leermakers FAM, Fleer GJ, Koopal LK, Zhulina EB, Borisov OV. Macromolecules 1999;32:2365e77. [26] Hariharan R, Biver C, Mays J, Russel WB. Macromolecules 1998;31:7506e13. [27] Harden JL, Cates ME. Phys Rev E 1996;53:3782e7. [28] Shaqfeh ESG, Doyle PS, Gast AP. Phys Rev Lett 1997;78:1182e5. [29] Grest GS. Phys Rev Lett 1996;76:4979e82. [30] Peters GH, Tildesley DJ. Phys Rev E 1995;52:5493e501. [31] Gutsche C, Salomo M, Kim YW, Netz RR, Kremer F. Microfluid Nanofluid 2006;2:381e6.

[32] Kegler K, Salomo M, Kremer F. Phys Rev Lett 2007;98:058304. [33] Kegler K, Konieczny M, Dominguez-Espinosa G, Gutsche C, Salomo M, Kremer F, et al. Phys Rev Lett 2008;100:118302. [34] Elmahdy MM, Synytska A, Drechsler A, Gutsche C, Uhlmann P, Stamm M, et al. Macromolecules 2009;42(22):9096e102. [35] Rühe J, Ballauff M, Biesalski M, Dziezok P, Gröhn F, Johannsmann D, et al. Polyelectrolyte brushes. In: Schmidt M, editor. Polyelectrolytes with defined molecular architecture I, advances in polymer Science, vol. 165. Berlin/Heidelberg: Springer; 2004. p. 189e98. [36] Prigogine I, Rice SA, editors. Advances in chemical physics, vol. 94. New York: Wiley; 1996. p. 219e331. [37] de Gennes PG. Adv Colloid Interface Sci 1987;27(3e4):189e209. [38] Liu W, Cholli AL, Nagarajan R, Kumar J, Tripathy S, Bruno FF, et al. Am Chem Soc 1999;121:11345e55. [39] Wang GM, Reid JC, Carberry DM, Williams DRM, Sevick EM, Evans DJ. Phys Rev E 2005;71:046142. [40] Peterman EJG, Gittes F, Schmidt CF. Biophys J 2003;84:1308e16. [41] Bouchiat C, Wang MD, Allemand JF, Strick T, Block SM, Croquette V. Biophys J 1999;76(1):409e13. [42] Odijk T. Colloids Surf A 2002;210:191e7. [43] Biver C, Hariharan R, Mays J, Russel WB. Macromolecules 1997;30:1787e92. [44] Waigh TA. Rep Prog Phys 2005;68:685e742. [45] Reichl LE. A modern course in statistical physics. 2nd ed. New York: WileyInterscience; 1998. [46] Doi M, Edwards SF. The theory of polymer dynamics. Oxford: Oxford University Press; 1992. [47] Faxén H. Ann Phys 1922;373(10):89e119. [48] Happel J, Brenner H. Low reynolds number hydrodynamics. 2nd ed. Boston: Kluwer Adacemic; 1983. p 553. [49] Weast RC. Handbook of chemistry and physics. 53rd ed. Cleveland: CRC Press; 1973. [50] Levenberg KQ. Appl Math 1944;2:164e8; Marquardt D. SIAM J Appl Math 1963;11:431e41. [51] Salomo M, Kroy K, Kegler K, Gutsche C, Struhalla M, Reinmuth J, et al. J Mol Biol 2006;359:769e76. [52] Wolf AV. Aqueous solutions and body fluids. their concentrative properties and conversion tables. hoeber medcial division. New York: Harper & Row; 1966. [53] Gutsche C, Kremer F, Krüger M, Rauscher M, Weber R, Harting J. J Chem Phys 2008;129:084902. [54] Dominguez-Espinosa G, Synytska A, Drechsler A, Gutsche C, Kegler K, Uhlmann P, et al. Polymer 2008;49:4802e7. [55] Neuman KC, Block SM. Opt Trapping Rev Sci Instrum 2004;75:2787e809. [56] Svoboda K, Block SM. Annu Rev Biophys Struct 1994;23:247e85.