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ONLINE: LETTERS
PUBLISHED 30 NOVEMBER 2008 DOI: 10.1038/NMAT2338
Probing interfacial equilibration in microsphere crystals formed by DNA-directed assembly
Anthony J. Kim1 *, Raynaldo Scarlett1 *, Paul L. Biancaniello2 , Talid Sinno1 and John C. Crocker1
DNA is the premier material for directing nanoscale self-assembly, having been used to produce many complex forms14 . Recently, DNA has been used to direct colloids5,6 and nanoparticles7,8 into novel crystalline structures, providing a potential route to fabricating meta-materials9 with unique optical properties. Although theory1012 has sought the crystal phases that minimize total free energy, kinetic barriers13 remain essentially unstudied. Here we study interfacial equilibration in a DNA-directed microsphere self-assembly system5,6,14 and carry out corresponding detailed simulations. We introduce a single-nucleotide difference in the DNA strands on two mixed microsphere species, which generates a free-energy penalty5,15,16 for inserting impurity spheres into a host sphere crystal, resulting in a reproducible segregation coefcient. Comparison with simulation reveals that, under our experimental conditions, particles can equilibrate only with a few nearest neighbours before burial by the growth front, posing a potential impediment to the growth of complex structures. In earlier studies, we showed that micrometre-sized polymer spheres with single-stranded DNA grafted on their surfaces can form large, close-packed colloidal crystals5,6 when the DNA strands hybridize to bridge them together. Within this interaction system, some local annealing is possible owing to the fact that bridge formation is a dynamic, reversible process. For crystallization to occur, the microspheres must be highly monodisperse (<4% standard deviation in diameter) and the DNA-induced sphere sphere binding energy, Eb , must be the proper strengthtoo strong and the particles bind strongly to assemble fractal aggregates, too weak and assembled structures dissociate. One feature of DNA-mediated interactions is that the computed binding energy depends exponentially on the free-energy change, G, for bridge formation (see Supplementary Information), as confirmed by direct interaction measurements5 . Owing to the strong temperature dependence of G for DNA hybridization15 , the corresponding temperature window for crystal formation17 is only 0.5 C wide. Within this temperature range, crystallites nucleate homogeneously in less than 24 h, and grow to a size of a few thousand microspheres within another 1224 h. To better understand the annealing and equilibration that takes place on the growing crystal interface, we designed a system expected to form a solid-solution alloy (Fig. 1). Specifically, we combined two populations of 0.98-m-diameter polymer spheres (carboxylate-modified polystyrene, Seradyn) that are essentially identical in their preparation18 and physical parameters, but that bear short grafted strands of single-stranded DNA whose sequences
differ at a single nucleotide-base location (Fig. 1). When additional linker DNA strands containing two complementary sequences are added to the solution, hybridization leads to the formation of DNA bridges between particles (Fig. 1b), which in turn give rise to a time-averaged attractive interaction with a 15 nm range5 . The difference in DNA sequence in the two populations of microspheres, A and B, decreases the AB bridge formation energy relative to an AA bridge by an amount G, which can be computed a priori from DNA thermodynamics15,16 . As the sphere binding energy is an exponential function of the hybridization free energy5 , the mismatch alters the relative sphere binding energies amongst the different populations according to
AA Eb = e( AB Eb G/kB T )
,
AA Eb = 2 , BB Eb
(1)
where kB is Boltzmanns constant and T is the absolute temperature. This result predicts that particle segregation should be rather sensitive to changes in DNA sequence. The typical G for a single nucleotide mismatch is 2 kB T relative to a WatsonCrick match16 , corresponding to a sphere binding-energy ratio of = e 2 7. We can find the minimum AA binding energy required for crystal AA stability, Eb = 3.75 kB T , through simulations described below. Thus, if B contains a single mismatch then the binding energy between AB sphere pairs would be seven times smaller than the AA binging energy, or less than 1 kB T . Presumably such B spheres would not bind the growing A crystal significantly, and would be completely excluded. This hypothesis was confirmed by experiment (data not shown). To obtain a smaller difference in sphere binding energy, and therefore produce solid-solution crystals, we place different mismatching bases on both the A and B particles; see Fig. 1c. One system we studied contained GG and GA mismatches on the A and B particles, respectively, and had the smallest accessible GGG/GA 0.22 kB T per bridge16 . In this case, equation (1) predicts = 1.25, corresponding to AA AB AA AA Eb = Eb Eb = (1 1/ )Eb = 0.79 kB T , assuming Eb = 4 kB T . As the pairwise interactions are additive, the energetic cost of inserting one B sphere into a close-packed (12-fold-coordinated) host crystal of A spheres would still be 12 Eb 10 kB T , presumably leading to nearly total B exclusion, at least in a fully equilibrated solid-solution crystal. In practice, the absence of solid diffusion in close-packed colloidal systems precludes such bulk equilibration. Instead, we might expect a smaller degree of segregation to occur owing to surface equilibration or kinetic limitations on the crystal interface.
of Chemical and Biomolecular Engineering, The University of Pennsylvania, 220 S. 33rd St. Philadelphia, Pennsylvania 19104, USA, of Physics and Astronomy, The University of Pennsylvania, 209 S. 33rd St. Philadelphia, Pennsylvania 19104, USA. *These authors contributed equally to this work. e-mail: jcrocker@seas.upenn.edu.
2 Department
1 Department
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NATURE MATERIALS | VOL 8 | JANUARY 2009 | www.nature.com/naturematerials
2009 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT2338
a
Colloidal crystallite
LETTERS
a
AGG /BGA
b
A
A
B
A 1 m
5 m
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AGG /BGA
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Grafted DNA
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AGATTGAACTTTAGAGAGAGATTGAACTTTAGAGA TCTAACTTGAAATGTCT TCTAACTTGAAATGTCT
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AGATTGAACTTTAGAGAGAGATTGAACTTTAGAGA TCTAACTTGAAATATCT TCTAACTTGAAATGTCT
Figure 1 | Polymer microspheres grafted with single-stranded DNA can form solid-solution colloidal crystals. a, A host crystal of A spheres containing a small number of impurity B spheres. b, The colloidal crystal is held together by dynamic DNA bridges, formed by DNA strands grafted to the spheres surfaces hybridizing with strands in solution (green) to form bridges. At the DNA densities of this experiment, several dozen grafted strands are available to form bridges, but only a few bridges would be present at any given time. Linkers hybridizing to two strands on the same sphere do not signicantly affect the interaction under these conditions. c, To penalize B-sphere insertion, their strands contain a GA base mismatch (boxed), which has a 0.22kB T -higher free energy than the GG mismatch on the A spheres (boxed).
5 m
Figure 2 | Varying the DNA sequence alters the experimental segregation coefcients of binary colloidal crystals. a,b, AGG /BGA crystals formed from a 50:50 A:B stoichiometry suspension, with 9% B substitution. c,d, AGG /BGA crystals formed from a 90:10 A:B stoichiometry suspension, with 1.5% B substitution. e,f, An AGG /BGT suspension at 50:50 A:B stoichiometry shows no detectable incorporation. All crystals shown after 72 h incubation at 42.6 C.
The sensitivity of cocrystallization to DNA sequence and hybridization energy is borne out by experiment. White-light and fluorescence images of solid-solution alloy crystallites are shown in Fig. 2. The substitution ratio B:A inside the crystallites can be measured accurately by directly counting the fluorescent B particles within the bottommost layers of several dozen crystallites. The index mismatch between the spheres and the water precludes reliable imaging of crystallite interiors, for example using confocal microscopy. The GG/GA system showed a finite substitution ratio, which was 0.092 0.009 for crystallites grown from a suspension with 50:50 A:B stoichiometry (Fig. 2a,b), and 0.0154 0.0025 for crystallites from a 90:10 A:B stoichiometry suspension (Fig. 2c,d). In both experiments, the segregation coefficient (defined as the ratio of the fraction of minority B particles in the crystal to the fluid) was consistent with the value kseg = 0.18 0.02. For a second system with GG and GT mismatches on the A and B spheres, respectively, with a larger G = 1.25kB T , no substituted microspheres were observed (Fig. 2e,f), corresponding to kseg < 103 . As expected, the measured segregation coefficient is significantly larger than the expected bulk equilibrium value, bulk kseg = exp(12 Eb /kB T ) 5 105 . In an interfacialequilibration situation, we might suppose that segregation is determined by pair interactions with a limited number, Neff , of nearest neighbours on the interface, where Neff is a function of the interfacial microstructure around the predominant growth sites. Specifically, we could suppose that fluid particles bind to and equilibrate with a few neighbouring particles on the
NATURE MATERIALS | VOL 8 | JANUARY 2009 | www.nature.com/naturematerials
interface (3 < Neff < 6) before being buried by a growth front, locking in the interfacial stoichiometry. Substituting Neff for the 12 in the preceding formula and rearranging, we find the experimental Neff ln(kseg )kB T / Eb = 1.90 0.35, assuming a reasonable binding energy and kseg = 0.18 (see Supplementary Information). The small value of Neff immediately suggests that growing DNA-directed particle assemblies incorporate defects even more easily than interfacial equilibration would suggest. Indeed, it is not clear how interfacial equilibration could operate with just two neighbouring particles or how a completely non-equilibrium kinetically controlled process could yield a reproducible segregation coefficient. Our experimental findings were compared with detailed, canonical-ensemble Metropolis Monte Carlo (MMC) simulations19 . A seed crystal of pure A (150 particles) was placed in a bulk fluid (5,000 particles) of a given A:B stoichiometry and the system allowed to evolve without constraint, during which the crystal growth rate and kseg were monitored. A sequence of MMC runs AA was performed using different values of Eb (3.75 kB T 5.0 kB T ), AB Eb (0.4 kB T < Eb < 1.5 kB T ), volume fraction (0.25 < < 0.4) and MMC step sizes; see Supplementary Information for details. Different simulation conditions led to large variations in segregation coefficient, crystal growth rates and bulk particle diffusion coefficients, DA and DB . Different runs were parameterized using a dimensionless crystal growth rate, D : the observed time-independent growth rate rescaled by the particles bulk fluid diffusivity, D diff /growth = (dr /ds) /DA , where is the particle diameter and r the crystal radius, and both the growth rate and the diffusivity, DA , are computed in units of MMC sweep number, s. Shown in Fig. 3 is a plot of Neff versus D
53
2009 Macmillan Publishers Limited. All rights reserved.
LETTERS
3 (ii)
NATURE MATERIALS DOI: 10.1038/NMAT2338
a
3 2 1
2
Neff
(i) 1
(iii)
Neff
0
102
101
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b
0 103 102 101 100
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Figure 3 | Simulation results show that segregation is controlled by the crystal growth rate. Specically, Neff collapses to a master curve when plotted against scaled growth rate, D diff /growth . Unlike the atomic case, segregation values show an unexpected staircase-like progression of plateaux; the dashed line is a guide for the eye. The diamond symbol corresponds to the experiment. (i)(iii) Simulated crystallites (all at AA 50:50 A:B stoichiometry suspension, = 0.3, B species in red): Eb = 3.75 kB T , AA AA Eb = 1.0 kB T (i); Eb = 4.25 kB T , Eb = 0.7 kB T (ii); Eb = 6.0 kB T , Eb = 0.4 kB T (iii). High growth rates (ii, iii) show essentially stoichiometric (50:50) substitution, unlike (i). For the fastest-growth case, crystal morphology becomes dendritic (iii), while remaining well ordered.
Neff
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for various simulation conditions. Notably, all the data points lie on a single master curve, suggesting that segregation behaviour is primarily controlled by a competition between growth and processes whose rates are proportional to diffusion. At high values of D (above 0.2), corresponding to the fastest growth dynamics, Neff 0, indicating fully non-equilibrium stoichiometric growth. In this limit, changing Eb (at fixed D ) does not significantly affect the composition of the growing crystal, and growth is still morphologically in complete equilibrium, that is, the seed growth remains perfectly crystalline with a roughly spherical shape (Fig. 3, (ii)). The onset of the dendritic shape instability2022 is seen by D 0.7 (Fig. 3, (iii)), although the cluster remains well ordered throughout. Non-stoichiometric substitution was observed in the simulations for lower values of D (Fig. 3, (i)). In the interval 0.05 < D < 0.2, Neff rises rapidly until a plateau is reached at Neff 2. This plateau extends across approximately one decade down to D 0.003, at which point Neff once again rises rapidly to a value of about 3. Taken together, our simulation results suggest the beginning of a staircase-like hierarchy of discrete segregation coefficients corresponding to different integer values of Neff (simulations at still lower growth rates were not practical). To make contact with the experimental data, we rely on our earlier finding6 that particle dissociation rates and crystal growth rates are close to those expected for a diffusion-limited case, provided the diffusivity, DA 0.03 m2 s1 , is corrected for lubrication effects23 . Along with the experimental crystal growth rate (3 104 m s1 ), we find an experimental D of 0.01, which places the experiment on the simulation master curve within the Neff 2 plateau, consistent with the experimental Neff = 1.90 0.35; see Fig. 3, diamond symbol. Moreover, the plateau-like nature of the simulated master curve explains why the experimental segregation coefficient was reproducible in the face of likely run-to-run variations in incubation temperature and particle density. Conversely, it was not possible to achieve a different segregation coefficient by varying the experimental D , owing to diffusion limitations and a finite lifetime (710 days) for the suspension to remain chemically stable.
54
Figure 4 | Different segregation steps correspond to the rates of particle escape from different congurations. ac, Simulation predictions of the Neff value versus growth rate rescaled by the escape rate from singly bound congurations (a), doubly bound congurations (b) and triply bound congurations (c). We nd that transitions between plateaux occur when the rescaled growth rates have a value of 0.3, corresponding to Neff increasing by 1 when the corresponding escape rate has become a few times faster than the rate of burial by the growth front.
An appealing explanation for the discrete segregation coefficients observed in the simulations would be if each plateau corresponded to a case where dissociation processes for particles with Neff neighbours were fast compared with the growth rate, whereas the dissociation of particles with Neff + 1 neighbours was too slow. To test this hypothesis, we estimated the time for a particle bonded to N neighbours to thermally dissociate and diffuse away from the crystallite:
n diss = AA L2 NEb (2 )2 W exp + , DA kB T DA
(2)
where LW is the range of the interaction potential. The first term in equation (2) approximates the timescale for bond breaking, and the second term approximates the time associated with diffusion across a boundary layer 2 thick. A sequence of renormalized growth n rates can now be defined, n diss /growth , which compare the relative rates of crystal growth to each escape process. The simulated segregation coefficients are replotted in Fig. 4 relative to these new scaled growth rates. This analysis confirms our explanation for the plateaux: in each case, the transition between plateaux occurs when the corresponding n value drops below unity. No plateau is 1 2 expected at Neff = 1, because diss diss owing to single-neighbour dissociation rates being fast relative to boundary-layer diffusion (see equation (2)), making 1 2 . The neat progression of segregation coefficients thus arises from the well-separated timescales for the dissociation rates of particles with different numbers of bonded neighbours. This in turn is a
NATURE MATERIALS | VOL 8 | JANUARY 2009 | www.nature.com/naturematerials
2009 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT2338
consequence of the strong, very short-ranged potentials in our system, characteristic of DNA-directed interactions. In contrast, the continuous segregation coefficients typically observed in atomic systems24 arise from longer-range potentials that efficiently funnel particles to the correct lattice site. In conclusion, our experimental system provides a probe of the interfacial equilibration processes that prevail during DNAdirected particle self-assembly. The capabilities of DNA-directed particle assembly lie in its ability to independently programme different-strength attractions among multiple particle species, in principle enabling the assembly of complex alloy and superlattice structures. We find surprisingly weak impurity segregation, even at growth rates where crystals are morphologically compact and well ordered, clearly demonstrating that morphological equilibration occurs at substantially faster timescales than chemical equilibration. At the very least, this suggests that multicomponent systems with small binding-energy differences may tend to a large number of substitution defects. In the worst-case scenario, we suppose that some energetically favourable structures may not be kinetically accessible, owing to inadequate surface equilibration causing particles to bind incorrectly, in turn generating a preponderance of structural defects. In our case, we have shown that our low segregation efficiency is a consequence of growth and segregation proceeding by the serial addition of single particles. A practical route to forming more highly ordered structures will probably be found by optimizing the form and range of the interaction potential, as well as by using hierarchical approaches, which first form small clusters that subsequently assemble into crystals13 . Recent experiments at the nanoparticle scale7 have shown that successful superlattice formation requires long DNA bridges, corresponding to a long interaction range. Formulations with shorter DNA spacers formed only amorphous structures, for reasons that are not understood, which we speculate may be related to kinetic effects at the interface. Last, our work demonstrates the utility of our quantitative interaction models and simulation framework for replicating the phase behaviour and growth kinetics of DNAdirected particle self-assembly.
LETTERS
7. Nykypanchuk, D., Maye, N. N., van der Lelie, D. & Gang, O. DNA-guided crystallization of colloidal nanoparticles. Nature 451, 549552 (2008). 8. Park, S. Y. et al. DNA-programmable nanoparticle crystallization. Nature 451, 553556 (2008). 9. Linden, S. et al. Magnetic response of metamaterials at 100 terahertz. Science 306, 13511353 (2004). 10. Tkachenko, A. V. Morphological diversity of DNA-colloidal self-assembly. Phys. Rev. Lett. 89, 148303 (2002). 11. Bozorgui, B. & Frenkel, D. Liquidvapour transition driven by bond disorder. Phys. Rev. Lett. 101, 045701 (2008). 12. Largo, J., Starr, F. W. & Sciortino, F. Self-assembling DNA dendrimers: A numerical study. Langmuir 23, 58965905 (2007). 13. Lukatsky, D. B., Mulder, B. M. & Frenkel, D. Designing ordered DNA-linked nanoparticle assemblies. J. Phys. Condens. Matter 18, S567S580 (2006). 14. Milam, V. T., Hiddessen, A. L., Crocker, J. C., Graves, D. J. & Hammer, D. A. DNA-driven assembly of bidisperse, micron-sized colloids. Langmuir 19, 1031710323 (2003). 15. SantaLucia, J. A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. Proc. Natl Acad. Sci. USA 95, 14601465 (1998). 16. Peyret, N., Seneviratne, P. A., Allawi, H. T. & SantaLucia, J. Nearest-neighbor thermodynamics and NMR of DNA sequences with internal A A, C C, G G, and T T mismatches. Biochemistry 38, 34683477 (1999). 17. Biancaniello, P. L., Crocker, J. C., Hammer, D. A. & Milam, V. T. DNA-mediated phase behavior of microsphere suspensions. Langmuir 23, 26882693 (2007). 18. Kim, A. J., Manoharan, V. N. & Crocker, J. C. Swelling-based method for preparing stable, functionalized polymer colloids. J. Am. Chem. Soc. 127, 15921593 (2005). 19. Auer, S. & Frenkel, D. Numerical simulation of crystal nucleation in colloids. Advanced computer simulation approaches for soft matter. Sciences I 173, 149208 (2005). 20. Mullins, W. W. & Sekerka, R. W. Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34, 323329 (1963). 21. Mullins, W. W. & Sekerka, R. W. Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35, 444451 (1964). 22. Johnson, B. K. & Sekerka, R. F. Diffusion-limited aggregation: Connection to a free-boundary problem and lattice anisotropy. Phys. Rev. E 52, 6404 (1995). 23. Biancaniello, P. L. & Crocker, J. C. Line optical tweezers instrument for measuring nanoscale interactions and kinetics. Rev. Sci. Instrum. 77, 113702 (2006). 24. Beatty, K. M. & Jackson, K. A. Monte Carlo modeling of dopant segregation. J. Cryst. Growth 271, 495512 (2004).
Acknowledgements
These studies were supported by the National Science Foundation under the DMR, NIRT and MRSEC programs. We thank A. Alsayed, Y. Han, V.N. Manoharan, V. T. Milam, M. Ung and M.-P. Valignat for discussions.
Received 24 April 2008; accepted 28 October 2008; published online 30 November 2008 References
1. Winfree, E., Liu, F. R., Wenzler, L. A. & Seeman, N. C. Design and self-assembly of two-dimensional DNA crystals. Nature 394, 539544 (1998). 2. Seeman, N. C. Biochemistry and structural DNA nanotechnology: An evolving symbiotic relationship. Biochemistry 42, 72597269 (2003). 3. Rothemund, P. W. K. et al. Design and characterization of programmable DNA nanotubes. J. Am. Chem. Soc. 126, 1634416352 (2004). 4. Rothemund, P. W. K. Folding DNA to create nanoscale shapes and patterns. Nature 440, 297302 (2006). 5. Biancaniello, P. L., Kim, A. J. & Crocker, J. C. Colloidal interactions and self-assembly using DNA hybridization. Phys. Rev. Lett. 94, 058302 (2005). 6. Kim, A. J., Biancaniello, P. L. & Crocker, J. C. Engineering DNA-mediated colloidal crystallization. Langmuir 22, 19912001 (2006).
Author contributions
The experiments were designed by A.J.K., P.L.B. and J.C.C., implemented by A.J.K., and analysed by A.J.K., T.S. and J.C.C. The simulations were designed by R.S. and T.S., implemented by R.S. and interpreted by R.S., J.C.C. and T.S. The interaction model was implemented by R.S. and P.L.B. J.C.C. and T.S. wrote the manuscript and oversaw the project.
Additional information
Supplementary Information accompanies this paper on www.nature.com/naturematerials. Reprints and permissions information is available online at http://npg.nature.com/ reprintsandpermissions. Correspondence and requests for materials should be addressed to J.C.C.
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Chapter1ManagerialAccountingand theBusinessEnvironmentSolutionstoQuestions11 Managerialaccountingisconcerned withprovidinginformationtomanagersforuse withintheorganization.Financialaccountingis concernedwithprovidinginformationto stockholders,creditors

UC Irvine - GEOG - 5B

1- Geography is the study of the evolving character and organization of the earths surface by natural and human factors/processes. - Evolving implies that the discipline is dynamic in that there are processes, which bring about changes. - Character is th

UC Irvine - BIO SCI - E109

Physiology Final ReviewLecture 21: Fluid & Electrolyte Balance Integrates several systems o Overlap in functions and so change made in one system affects others o Decrease in blood volume and blood pressure triggers homeostatic reflexes in cardiovascula

North South University - MBA - BUS 511

McClave - Statistics 11e - Chapter 1

UCSD - BIMM - 100

UCLA - EE - 110

Imaginary NumberCan we always find roots for a polynomial? The equation x2 + 1 = 0 has no solution for x in the set of real numbers. If we define a number that satisfies the equation x 2 = -1 that is, x = - 1 then we can always find the n roots of a poly

UCLA - EE - 110

UCLA - EE - 110

UCLA - EE - 110

UCLA - EE - 110

ReviewClass#1ReviewsonEE10EE100,Spring2009 UniversityofCalifornia,LosAngelesTA:LokwonKim(Ph.D.inEE)ResistanceResistanceacharacteristicbehaviorof resistingtheflowofelectricchargeorability toresistcurrentOhmsLawOhmsLawThevoltageV acrossaresistorisdir

UCLA - EE - 110

TRIGONOMETRYAngles, Arc length, ConversionsAngle measured in standard position. Initial side is the positive x axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive,

UCLA - EE - 110

EE110 HW3Due: 4/27 (Tues) before classPractice ProblemsPractice Problem 1.Practice Problem 2.Practice Problem 3 Find the Thevenin equivalent circuit for Fig. below.Graded ProblemsProblem 1 (10 pts)Problem 2 (10pts)Problem 3 (20 pts) To get full c

UCLA - EE - 110

UCLA - EE - 110

UCLA - EE - 110

Question 1:Question 2:Question 3

UCLA - EE - 110

Homework 8Practice ProblemsGraded ProblemsSolutions to Practice Problems

UCLA - EE - 110

Question 3

UCLA - EE - 110

EE110 Homework-2: RLC Circuits Practice ProblemsPractice Problem 1Practice Problem 2Graded ProblemsProblem 1: (10 pts)Problem 2: (15pts)Problem 3: (15pts)

UCLA - EE - 110

UCLA - EE - 110

EE110: HOMEWORK 4Practice Problem 1Practice Problem 2Practice Problem 3:Practice Problem 4:Practice Problem 5:Graded Problems: Problem 1:Problem 2:Also, plot the power dissipated by RL by varying RL using PSPICE. Verify that the power reaches its

UCLA - EE - 110

Hw4: Practice Problem SolutionsProblem 3:Problem 4:Problem 5:

UCLA - EE - 110

Homework 4: SolutionsQuestion 1Question 2Question 3Question 4

UCLA - EE - 110

EE110 HW5Due: 5/11 TuesdayPractice ProblemsPractice Problem 1.Practice Problem 2.Determine the one-sided Laplace transform ofPractice Problem 3Practice Problem 4Given L1 = 1 H and C = 1/2 F. Find i(t) and vc(t) for the circuit shown below, when vc

UCLA - EE - 110

UCLA - EE - 110

EE110 HW6 Practice ProblemsPractice Problem 1.Find v2(t) in the circuit below.Due:5/18 TuesdayPractice Problem 2.Find v0(t) for circuits (a) & (b), and find i0(t) for (c) & (d).Practice Problem 3.Practice Problem 4Practice Problem 5Obtain H(s) =

UCLA - EE - 110

UCLA - EE - 110

Homework 7Practice Problem 1:Practice Problem 2:Practice Problem 3:Practice Problem 4:Graded Problem 1:Graded Problem 2:Graded Problem 3: Use a 500nF capacitor to design a band-reject filter, as shown in the figure. The filter has a center frequenc

UCLA - EE - 110

Hw7: Practice Problem SolutionsProblem 1Problem 2:Problem 3Problem 4:

UCLA - EE - 110

Question 1: Alternate Solution