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An investigation of the aggregation behavior of carnation ring spot virus using dynamic light scattering Williams, Lloyd 1986

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A N OF  I N V E S T I G A T I O N C A R N A T I O N  OF  R I N G L I G H T  T H E  A G G R E G A T I O N  S P O T  V I R U S U S I N G  B E H A V I O R D Y N A M I C  S C A T T E R I N G .  by  LLOYD WILLIAMS B. Sc., Imperial College, London, 1983  A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in  The Faculty of Graduate Studies Department of Physics  We accept this thesis as conforming to the required standard  The University of British Columbia April 1986 ©  L L O Y D W I L L I A M S , 1986  jg>-  In  presenting  requirements  this for  an  of  British  it  freely available  agree for  that  I  by  understood  that  his  copying  f i n a n c i a l gain  DE-6  (3/81)  University shall  reference  and  study.  I  extensive be  her or  shall  copying  granted  by  the  of  publication  not  be  allowed  Columbia  of  make  further this  head  representatives.  of  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l Vancouver, Canada V6T 1Y3  Date  the  Library  permission.  Department  at  the  the  may  or  f u l f i l m e n t of  that  for  purposes  or  degree  agree  for  permission  scholarly  in partial  advanced  Columbia,  department  for  thesis  It  this  without  thesis  of  my  is  thesis my  written  ABSTRACT  The technique of digital autocorrelation of intensity fluctuations of scattered laser light is used to investigate the aggregation behavior of carnation ringspot virus. It is hypothesized by the author that the aggregation rate (R) is given by  exp\-E /k {T-T )] agg  B  c  \iT>T  c  if r <T  C  R is measured for a range of values of the dependent experimental variables, temperature (T), and virus concentration (i/o). And the dependence of the two aggregation parameters, the temperature of aggregation (7^) and the energy of aggregation {Eagg),  on the experimental variables is ascertained.  Also certain physical properties of the virus are measured: size, molecular weight, and diffusion coefficient. The experimental accuracy of the apparatus is determined by performing a series of experiments on a known system of Latex spheres.  iii  TABLE OF  CONTENTS  ABSTRACT  ii  TABLE OF CONTENTS  iii  LIST O F T A B L E S  v  LIST O F F I G U R E S  vi  ACKNOWLEDGMENTS CHAPTER  1: I N T R O D U C T I O N  C H A P T E R 2: T H E O R Y  viii 1 4  21 Introduction  4  2-2 Synopsis  5  2-3 The Theory of Light Beating Spectroscopy  7  2-4 Light Scattering From Macromolecules  10  2-5 Calculation of the Self Intermediate Scattering Function  16  2-6 Koppel Method of Cumulants  19  2- 7 Theory of Brownian Aggregation  21  C H A P T E R 3: E X P E R I M E N T  31  31 The Correlator  31  3- 2 The Digital Correlator Computer Interface  34  3-3 The Index Matched Cell and Temperature Control Device  36  3-4 The Optics  37  3-5 The Preparation of the Sample  39  3- 6 The Experimental Procedure  40  C H A P T E R 4: D A T A A N A L Y S I S  43  41 Monodispersive Data Analysis  43  4- 2 Polydispersive Data Analysis  45  CHAPTER  5: R E S U L T S  51 Introduction  54 54  IV  5-2 Latex Sphere Results  54  5-3 CRSV Results -I- Nonaggregating Regime  60  5-4 CRSV Results -II- Aggregating Regime  64  5-5 Suggestions for Further Study  73  C H A P T E R 6: C O N C L U S I O N S  74  BIBLIOGRAPHY  75  A P P E N D I X 1: E V A L U A T I O N O F V E L O C I T Y A U T O C O R R E L A T I O N FUNCTION  77  A P P E N D I X 2: C O R R E L A T O R C O N T R O L S  80  A P P E N D I X 3: L I S T I N G O F C O R R . A G  84  V  LIST OF TABLES  V - l Table of the results of measurements on Latex spheres  55  V-Il Table of the results of measurements on CRSV  60  V-1I1 Values of the aggregation energy and temperature of aggregation  69  vi  LIST OF FIGURES  1- 1 Effect of temperature on two strains of CRSV  3  2- 1 Schematic representation of the light scattering experiment  5  2-2 Schematic representation of the light scattering experiment  6  2- 3 Scattering vector geometry  12  3- 1 Digitization of fluctuating light intensity  32  3-2 Schematics of a digital clipped correlator  33  3-3 Digital correlator interface block diag  35  3-4 Index matched cell  37  3- 5 A diagram of the apparatus  38  4- 1 CORR.AG output Pg.l  47  4-2 CORR.AG output Pg.2 .  48  4-3 CORR.AG output Pg.3  49  4-4 CORR.AG output Pg.4  50  4- 5 Average temperature of cell once in I.M.C  52  2  5- 1 Variation in f as a function of g  56  5-2 Variation in f as a function of q  57  5-3 Variation in f, for CRSV, as a function of temperature  61  5-4 Variation in f, for CRSV, as a function of time  63  5-5 Theoretical fit by ANALYSIS to aggregation data  66  5-6 Alternative fit by ANALYSIS to aggregation data  67  5-7 Final fit by ANALYSIS to aggregation data  68  5-8 Log of the aggregation rate as a function of temperature  70  5-9 Energy of aggregation as a function of concentration  71  5-10 Temperature of aggregation as a function of concentration  72  Al-1 Correlator front panel  82  ACKNOWLEDGMENTS It has been a pleasure to work in the U.B.C. physics department critical phenomena lab. I would like to thank Dr. David Balzarini for his assistance and guidance, and Jack Tremaine and Bill Ronald of the Vancouver Agriculture Canada Station, for providing me with the virus samples and advice on the virological aspects of the experiment. 1 am also greatly indebted to John de Bruyn for many helpful discussions, and to Ulrike Narger for her co-operation in sharing dark-room facilities. I would also like to thank Alex Harper for proof reading this thesis, and Mary-Anne Potts for writing the original version of COR.AG—CORR.S. Finally, and most importantly, I wish to thank Urmil whose support and encoragement was of great value to me.  CHAPTER 1: INTRODUCTION  CHAPTER 1 INTRODUCTION  The intention of this thesis is to present the results of an investigation of 1  (CRSV): size, molecular  weight, diffusion coefficient, and aggregation behavior.  A n approach based on  certain physical properties of carnation ringspot virus  2  Smoluchowski's theory of Brownian aggregation is used to model the aggregation phenomenon. This is combined with a temperature dependent probability of aggregation that is hypothesized by the author. As a result a variable can be defined which characterizes the rate of aggregation. This is then measured for various combinations of the experimental variables, temperature and concentration, allowing the dependence on concentration of the aggregation parameters to be determined. Light scattering spectroscopy was the experimental technique used. Its capability to make rapid and accurate measurements of particle size, diffusion coefficient, and polydispersity make it an ideal tool for such an investigation. Prior to the study of C R S V , a control system of polystyrene latex spheres was examined to ascertain the characteristics and limits of accuracy of the apparatus. Samples of Carnation Ring Spot Virus were provided by Dr J . H . Tremaine of the Agriculture Canada research station in Vancouver. There are three reported strains of CRSV: N , A , and R. These three strains show no differences in host reactions, but do differ in such properties as particle aggregation, amino acid composition and serological properties.  Particles are stabilized by a pH dependent  1  CHAPTER  1: INTRODUCTION  protein-protein interaction, and are stored in a pH 5 sodium acetate buffer to prevent swelling. Particles of the A strain can form six-particle aggregates, and linked aggregates. Particles of the R and N strain aggregate in a temperature-reversible 8  4,5  manner: first investigated by J.H.Tremaine and W.P.Ronald ' . The purpose of this study is to investigate the behavior of CRSV-N. Experiments by Tremaine and Ronald have consisted of measuring the absorbance of virus preparations in different concentrations, at a wavelength of 340nm, as the samples were heated at a rate of 0-25 °C per min. Some of their results* are shown in Fig. 1-1. It can be seen that the temperature required to induce aggregation in each of the strains increases with decreasing virus concentration. It was also observed that the maximum absorbance on heating CRSV was directly proportional 4  to the logarithm of the virus concentration. Tremaine and Ronald also raised the question of whether the molecular weight of the aggregates formed at maximum absorbance was similar for all concentrations. The use of light scattering spectroscopy has the advantage over absorbance studies that it directly measures the particle size. It also provides information as to whether the viruses are still aggregating once the system has become turbid. This is when the extinction coefficient as measured by the absorbance studies has become a constant. The process of aggregation is endothermic, and therefore driven by entropy. It is also reversible. It is thought that hydrophobic bonds along with conformational changes in the protein are involved in the aggregation process.  2  CHAPTER  1: INTRODUCTION  Figure 1-1 Effect of temperature on two strains o f CRSV at various concentrations (measured in mg/ml) illustrating the increase in absorbance due to aggregation caused by heating.  3  CHAPTER 2: THEORY  CHAPTER 2 THEORY  2.1 Introduction The technique of digital autocorrelation of intensity fluctuations of scattered laser light has been shown to afford precise and rapid measurements of the transla6,7  tional diffusion constants of macromolecules . This information can be used, with the Einstein relation for the diffusion constant, to give the hydrodynamic radius for the molecule. If this is combined with the Koppel method of cumulants approach  8  to an analysis of the correlation function, information on the polydispersity of the sample can be obtained. This technique can then be used to investigate the aggregation properties of a system, as reflected in the increase in the mean radius and polydispersity of the scattering particles in that system. Chapter 2 will begin with a synoptic view of light scattering experiments. The theory of light beating spectroscopy is then discussed in terms of the relationship between the optical spectrum of light scattered by macromolecules, and the spectrum of the measured photoelectric current. In the next section the concept of a self-intermediate scattering function will be introduced, and its relationship to the optical spectrum of a dilute system of macromolecules will be examined. A derivation of the self-intermediate scattering function of such a system will then be undertaken. This will yield an analytical expression describing the spectrum and corresponding correlation function of a monodispersive system. The Koppel method  4  CHAPTER 2. THEORY  of cumulants approach to the analysis of correlation data will then be outlined, and and an analytical expression for the correlation function of a polydispersive system will be derived. The last section will review the theory of Brownian aggregation. 2.2 Synopsis In a light scattering experiment, light which has been passed through a polarizer to define the polarization of the incident beam impinges on a scattering medium. The scattered light then passes through an analyser which selects a certain polarization and finally enters a detector.  The position of the detector defines  the scattering angle 0. In addition, the intersection of the incident beam and the beam intercepted by the detector defines a scattering region of volume V. This is illustrated in Fig. 2-1. SCATTERING VOLUME  LASER k,»  POLARIZER  DET ECTOR  Figure 2 - 1 Schematic representation of the light-scattering experiment.  2rn  b  CHAPTER  2: THEORY  In modern light scattering experiments a photomultiplier is the main detector, but the pre- and post- photomultiplier systems will differ according to the frequency spectrum of the scattered light. The three different methods- filter, homodyne (or self-beat), and heterodyne are schematically illustrated in Fig 2-2.  INTERFEROMETER  o)  SCATTERED LIGHT  " DIFFRACTION  b)  RECORDER  ORATING  PHOTOMW.TIPLER SCATTERED  PHOTOMULTIPLIER  •r  AUTOCORRELATOR  ,  LIGHT  •r SPECTRUM  PHOTOMULTIPLER  ANALYSER  AUTOCORRELATOR •r S P E C T R U M ANALYZER  RECORDER •r PRINTER  RECORDER •*  PRINTER  F i g u r e 2-2 Schematic illustration of the various techniques used in light-scattering experiments: a) filter methods; b) homodyne; c) heterodyne.  The  spectral characteristics of the scattered light depend on the time scales  characterizing the motions of the scatterers. The quantities measured in light scattering experiments are derived from the time correlation functions of either the scattered electromagnetic field or the scattered intensity. Consequently the core of this chapter is devoted to the derivation of an analytical expression that describes the correlation function of the intensity of light scattered by a dilute system of spherical particles. This is an adequate model for the system in the experiment.  6  CHAPTER 2: THEORY  Since certain time scales crucial to the understanding of this subject are mentioned during this chapter it will be instructive to explain them at the outset. T is defined T  as the relaxation time of the field correlation function. r is defined as the charf  acteristic time of the velocity autocorrelation function of the scattering particle. In other words, the 'memory' of the molecule's velocity during a given interval of time—say between t and t + dt—is 'completely lost' after a lapse of time large in comparison with r . Likewise t* is defined as the characteristic correlation time for a  the fluctuating force on the scattering particles that arises from the incessant and random bombardment of the solvent molecules. It is found that T » t, T  t* .  2.3 The Theory of Light Beating Spectroscopy Consider a classical coherent field at the photocathode of the photomultiplier specified by the vector E(t) = E exp(-*"u;r). 0  (2-1)  The probability per unit time of photoelectron emission from the photocathode is found to be =  oE'{t)E{t),  (2-2)  where a is a the quantum efficiency. The corresponding photocurrent is  i{t) = eWW{t) = toE'(t)E(t).  (2-3)  9  The Wiener-Khinchin theorem expresses the power spectrum of the photocurrent as (2-4) — 00  7  CHAPTER  S: THEORY  where C,(r)(the current autocorrelation function) is  2  Ci{r) = {i{t)i(t + r)) = e (w^{t)W^(t  + r)) .  (2-5)  The photocurrent actually consists of a series of discrete pulses. Therefore Ci(r)  has two distinct contributions: if the electrons at t and t + t are distinct,  (wW{t)wM{t  + r))  J  = (wl )(M + r)) 2  = " (/>V  2 )  (0,  (2-6)  where  (I) = (E'(t)E(t))  l«  ( r )  (2-7)  _ (S'(Qg(OE'(t + T)E(t + r))  while if the same electron occurs at t and t + r  ll,  (lvW(0W (* + r ) ) = ( w  (1,  ( 0 ) 6{t)  =o(I)6{t).  (2-9)  Therefore  J  C (r)=e a(/)^( ) + e V ( / ) V i  r  =e(i)6{T)  2 ,  (r)  2  + (i) gW(T).  (2-10)  The two commonly used detection schemes are the homodyne and heterodyne schemes. In the heterodyne scheme the light scattered from the system of interest is mixed with a local oscillator, whereas in the homodyne scheme—which is used in this experiment—only the scattered light impinges on the photocathode.  8  CHAPTER 2: THEORY  The electromagnetic field scattered by the system is characterized by the autocorrelation function  G^(r)  = (E'(t)E{t  +  r))=(I)g^(r).  (2-11)  1  The electromagnetic field correlation function j ^ ' is related to the intensity corre2  lation function j ^ ' , for random Gaussian fields, by the Siegart relation  9  {2)  (') = l + l 9 ( ' ) | U )  (2 - 12)  2  Accordingly, the current autocorrelation function becomes  C (r) i  = e(i)6{r)  2  + (i)  [l + \gW(r)?]  .  (2-13)  The experiment was performed with a digital correlator that uses the method of photocount autocorrelation in which the current is sampled at discrete intervals of time t, and recorded as a number of photons n{t ) t  interval t centred at time  detected in the sampling  Accordingly, (i(t)i(t — t)) would be constructed as a  sum of products N X] ( ' i M ' i - m ) i n  for a range of delay times mT  B  (2-14)  where m = 1 , 2 , 3 . . . M . M is the total number of  channels built into the autocorrelator, N is the number of sample times, and T, is the autocorrelator sample time. In practice, computation of Eq. 2-14 requires an excessive amount of digital electronics for the data acquisition and multiplication processes. A method of clipping the signal is used, therefore, in which a clipped count njt(fj is measured so that if n[ti) > k if!»(«,-) < k  9  CHAPTER 2: THEORY  where the non-negative integer k is the clipping level. A single clipped correlator then measures N ]T n(t,-)n*(t,-_ ).  (2-15)  m  The advantage of clipped over full correlation is that it allows the past history of photocounts to be stored as a linear chain of l's and O's. This leads to a considerable simplification in the circuitry of the correlator, with little loss of physical information. The single clipped correlation function is given b y  C {r) = ( n ( t , ) n ( t , _ ) ) = (n) {n ) 1 + lk  f c  m  k  1 0 , 1 1  (1 + fc) P\9(mT)\ (! + (»»  (2 - 16)  with \ ( T\\a  l  g  m  [  m  T  ^  =  (E'itW —  m  + mT.)) m  —  (2 - 17)  '  The factor 0 in E q . 2-16 takes into account such effects as the incomplete spatial 11  coherence of the light over a finite size detector and detector dark current . It is a complicated function of (n), k,mT , and the geometrical arrangement of the optical e  components. However, it is not seen to be a function of m for the correlation functions encountered in the experiment, but can be regarded as a constant parameter to be determined experimentally for a given experimental run.  2.4 L i g h t Scattering F r o m Macromolecules First let us consider what happens when an incident monochromatic beam described by E,(r,rJ = n , E e x p » [ k , - r 0  (2-18)  CHAPTER  2: THEORY  impinges on a single molecule which has a polarizability specified by the tensor a. The incident light will induce a dipole moment n given by  V{t)  (2-19)  = a -Ei{t).  If the fluctuation of the dipole moment is given by  tt(t)  —  ao + S a ( r , t ) ,  the electric  field of the scattered light at the detector, at position R with polarization Uj and 12  frequency ojf i s , E = —^-expikfR Rt  f I J  0  Eg(R,t)  0  expt(q • r -  iujt)  v [n/ • [k, x (k, x (6a[ ,t) T  • m))]] dr,  (2 - 20)  where the subscript V indicates that the integral is taken over the scattering volume. The vector q is defined in terms of the scattering geometry as  q = ki-k/.  (2-21)  Which for quasi-elastic light scattering i.e., jk,| = |kj|, is  Ann  ? = —  0  s»»  (2-22)  where n is the refractive index of the medium. This is illustrated in Fig. 2-3. Eq. 2-15 can be expanded in terms of spatial Fourier transforms of the dielectric fluctuations, and the vector cross products can be evaluated to give  — k? E —^expi{k R  E*{R,t)=  - uJit)6a {q,t),  f  i f  (2-23)  where 6a [q t) if  t  = n 6a{q,t) r  • n,  (2-24)  11  CHAPTER  2: THEORY  DETECTOR  F i g u r e 2-3 Light of polarization n, and wave vector k, is scattered in all directions. Only scattered light of wave vector k^ and polarization n/ arrives at the detector. The scattering vector q = k, - kf is defined by the geometry. Since the scattered wave has essentially the same wavelength as the incident wave kj ~ (27rn)/A, = fc,, it follows from the law of cosines that q = 2fc, sin 9/2  is the component of the dielectric constant fluctuation tensor along the initial and final polarization directions. For our purposes we are interested in the time correlation function k*\E I  2  (E;(R,0)E,(R,t))  = -^-^a, (q,0)^a, (q,0)exp-»u;,<, /  /  (2-25)  which we shall herein refer to as G^(t). By application of the Wiener-Khinchin  CHAPTER 2: THEORY theorem, the spectral density of the light scattered into the detector is given by  I {q,u ,R) if  f  = — "  Iif(q,ojf,R)  ^ 1  1  dt(6a {q,0)Sa {q,t))expi{u -u; )t.  /  if  if  f  i  (2-26)  * o J - oo  is therefore proportional to the spectral density of the fluctuations of  the molecular dipole moments I°j(q,uj)  :  I {q,u> ,R) if  oc /"/(q.w),  f  (2-27)  w here a  /,/(q,0 =(6a; (q,0)6a {q,t)), f  if  (2 - 28)  N  «a,-/(q,t) = $ 1 «?/(*) « p iq-r(t),  (2-29)  N  and  6a {T,t)=^a' {r-T{t)). lf  (2-30)  if  i=i It should be noted that this molecular approach to light scattering theory is clearly an approximation as it ignores collision induced effects. However, these effects are temporally separable from the motions of the molecules that are responsible for Rayleigh scattering, and thus only give insignificant unresolvable contributions to the correlation functions. For the purpose of this study our system can be modeled as spherical molecules with isotropic polarizabilites. Thus the tensor a becomes proportional to al where / is the 3 x 3 indentity matrix and a is a scalar. The elements a,y(q,<) in E q . 2-28-30 can be written as  «»/ = n, •  a  • n = a(n, • u ), ;  f  (2-31)  13  CHAPTER 2: THEORY  14  and N  £a,/(q,0 = (n,-  exp i  (q-ry(O) .  (2-32)  3=1  1  Thus G ^ ) is proportional to .Fi(q,r) where  *i(q.O = ^Eexp.q - [r,(0 -  rj  {t)]j  (2 - 33)  is the dynamic structure factor of the system. It should be made clear at this point that the summation is restricted to molecules in the scattering volume V . To emphasize this point we can introduce the following quantity for the j  th  particle :  Thus  *i(q,0= ( E MoJMOexptq-MO-Mo)]).  (2-34)  \t,,=i  The summation now extends over all N molecules in the system. In a typical solution of macromolecular Brownian particles it is found that the polarizability of the macromolecules is much greater than that of the solvent molecules and therefore they scatter more light.  The macromolecules also move  much more slowly than the solvent molecules, and consequently contribute a slowly fluctuating field at the detector as compared to the solvent molecules. The macromolecular motion should therefore be temporally separable from the solvent motion. And as macromolecules dominate the long time behavior, we need to only sum over these molecules.  CHAPTER  2: THEORY  In a sufficiently dilute solutions the macromolecules so rarely encounter each other we can assume their positions to be statistically independent. In this case Eq. 2-34 simplifies to  * i (q. 0 = (|e M ° ) M O  e x  P *'q • [/(0 - ;(°)] ^ • r  r  (2-35)  This is an example of a self-correlation function. 6  As the time scale for variation of bj{0)bj(t) is typically a factor of 10 longer than that of the argument of the exponent, it is permissible to set bj(0) bj(t) equal to its initial value 6y(0)6y(0). This is because 6/(0) can only have two values 0, or 1, 6j(0) = fc (0),and J  *i(q,0  e  = ^£M°) *p»'q- M O - M o ) ] ^ -  (2-36)  The quantity exptq- [ry(0 — ^(0)] is statistically independent of whether the particle j is in the scattering volume or not. Thus  N  *i(q>0 = E  ex  <M°)> ( P»'q •  h(0 -  2  37  (-)  The quantity F,(q,t)  = (exp.q • [r (r) - r (0)]) , y  y  the dynamic single particle structure factor of the system or the  (2 - 38)  self-intermediate  12  scattering function (S.I.S.F) as it is often called , is identical for each particle as it represents an ensemble average. F (q,t) s  can therefore be factored out of the above  15  CHAPTER  I" sum.  2: THEORY  \  Moreover, ( ]^&;(0) ) is simply (N) (the average number of particles in V)  so that F i ( q , r) becomes  F,(q,0 = (N)F,{q,t),:  (2-39)  •the mean number of particles in V times the S.I.S.F. To obtain an analytical expression for the correlation function, therefore, the S.I.S.F. for the system must be calculated.  2.5 C a l c u l a t i o n of the Self-Intermediate Scattering F u n c t i o n To calculate F {q, t) we first note that the argument of the exponential can e  be expressed as r(t) - r(0) = / v(r)dr. Jo  (2 - 40)  If q is taken in the x direction E q . 2-38 can then be written as  F,[q,t) = (  e x  P « 9 ^ v {r)dT^  (2-41)  x  where V (T) is the x component of v. If the exponential in Eq. 2-41 is then expanded, X  we get  F,{q,t) = 1 + ik I Jo 2  fc - 7 /  {v {r))dT x  /•'/•' /  (VZ{TI)V {TI)) X  dn  dr  2  * JO •'O l  +  ,-fc3 rt [ ft TT / / / { x{n)v {T2)v {T )) o! JQ JQ JQ v  X  - j j j J  Q  J  Q  J  x  Z  r  Q  dT dT dT X  2  Z  r  v M i ) v , ( r ) y , ( r s W « ) } d dT dT du. 2  Tl  2  Z  (2-42)  CHAPTER 2: THEORY  This expression can then be simplified by employing the following two assumptions: firstly, that the randomly fluctuating particle velocity v(t) has a mean value of zero and that the higher odd moments are all zero; secondly, that due to random collisions between the particles and the solvent molecules v(i) will change rapidly in magnitude and direction over the time scale of interest. If we denote the autocorrelation function of the particle's velocity K [ti v  K {h,h) v  ,t ) as 2  = (v[ti)v{t )),  (2-43)  2  a consequence of the second assumption is that  K (t t ) v  u  2  = 0 for | d - t | » r „ 3  where r is the characteristic relaxation time of the velocity autocorrelation function, 8  and is seen to be much smaller than the relaxation time of the electricfield(TV). Applying these assumptions to E q . 2-42 we see that from the first assumption all the terms which contain an odd power of k are zero; and from second assumption that the multiple integrals corresponding to the higher moments are only non-zero when pairs of times lie within the characteristic time for the velocity autocorrelation function. Thus, for example, in the k* term the integral can be split into three pairs of integrals, one corresponding to t\ w t and<3 w tt, another to t\ as t$ and t w ti, 2  2  and the third to ti w t< and t « h. Eq. 2-41 then reduces to 2  F.(q,t) = 1 - y  ^2TJ  (v (0)v (t)) dt x  x  +. oo  =  1  +  E  \ ^ j r (  TTl — \  2  m  - l)(2m-3)...5-31 [2r^  (v (0)v (t)) dt z  x  CHAPTER 2: THEORY 2  expj-fc r^  (v {0)v {t)) d*j x  (2-44)  x  The diffusion coefficient of the system can consequently be identified * as  D=  f Jo  (v {0)v {t)) x  dt.  x  (2-45)  The substitution of Eq. 2-45 into Eq. 2-44 gives  F,{q,t)  2  =  exp{-Dq T),  (2 - 46)  and E q . 2-16 becomes,  C (r) ik  =  [1 + k) „  {nl(n )  ,  (2 - 47)  k  The relaxation time of the field autocorrelation function can therefore be defined as  (2-48).  The reciprocal of TV is known as the Rayleigh linewidth (T) of the corresponding spectrum. Computer analysis of the correlation data can then yield the diffusion constant if the data are fitted to the function  C,-k(r) = (n)(n ) (1+7 e x p - 2 r r ] , fc  (2-49)  where T and 7 are treated as constant experimental parameters in a data fit. If the hydrodynamic diameter of the scattering particles (f) is desired, this can be obtained by utilization of the Einstein relation for the diffusion coefficient  D = * r/3iriK, B  * see appendix A  18  CHAPTER  2: THEORY  where 77 is the viscosity of the solvent. Hence  kT B  ,,  (»vr).  (2-50)  2.6 Koppel Method of Cumulants. For samples in which there is significant aggregation there can be a great diversity in the diameters of the scattering particles, which means that the correlation function can no longer be described by a single exponential. Aggregates of mean diameter f, will have a diffusion coefficient Z?, = k^T/GnT]^ and a corresponding relaxation parameter T . Consequently, the distribution of particle aggregates t  of various sizes in a polydispersive sample can be characterized by a distribution function P{T). Thus the correlation function is given by  {l)  \9 (r)\=  /°°P(r)exp(-rr)dr.  (2-51)  Jo  The remainder of this section illustrates how the correlation can be expressed in terms of the moments of the distribution function .P(r). It is noted the correlation function obtained from a polydispersive solution corresponds to the moment generating function K(-r;r)s(exp(-rr)>  (2-52) (2-53)  ll)  = \g (r)\.  This suggests analysis in terms of moments or cumulants.  The moments of the  distribution are related to the derivatives of ^ ( - r ; T ) with respect to  ^(r) =  (—t)  ( O  = [d /d(-r) ]K(-r;r)|_ m  m  r = 0  (2-54)  19  CHAPTER  2: THEORY  Similarly, one can define the cumulant generating function as the natural logarithm of the moment generating function:  *(-r;r)=ln«(-r;r) = ln  (2 - 55)  [1)  \g (r)\,  (2 - 56)  which thus takes the form of a power series in r. That is,  *(-r r)  *m(r) =  (2 - 57)  ;  •r=0  and (2 - 58) m=l  The cumulants can be written explicitly in terms of the moments  1=1  {  a  )  |Mi(r)r-|M«(r)]-  (2 - 59)  where c[l\ai •••a ) — m  ml  The sum over {a} includes the set of non negative integers a\... a  E  (2 - 60)  nr-i«yW''  a  3i  ~  m  '  m  for which  (2-61)  m  and  ^  a.j = I.  (2 - 62)  20  CHAPTER 2: THEORY  Accordingly, by combining Eq. 2-56 and 59 the cumulant generating function can be written as a series expansion in terms of the moments. Therefore:  i . , <.) |. - , + s f c g f  (r)  2ir For  +  r  2  3ir  3  '  „.«, 4ir  4  a single exponential correlation function Eq. 2-63 is linear in t. The power of  this approach derives from the fact that for many ff^(r)'s of interest the terms in Eq. 2-63 fall off rapidly with increasing order of tr. In the current analysis only the first two terms were kept. Consequently, the correlation function for polydispersive systems is  dk{r)=  (2 - 64)  (n)(n ) k  2  r can be related to an average diffusion coefficient. Assuming Tj = Djq , we get D = r/  2 9  2Z{h)Dj  (2 - 65)  3  E</>) 3  A quality parameter Q can also be defined as  2  66  ' " > Only if Q is zero to within experimental error can a system be considered monodispersive.  2.7 Theory of Brownian Aggregation The following approach to be discussed was first considered by Smoluchowski and  has been reviewed by S. Chandrasekhar  1 S  .  2  In this approach the system is  21  CHAPTER  2:  THEORY  supposed to consist initially of single particles that describe Brownian motion at a given rate determined by temperature, their size, and the viscosity of the solution that they are in.  Each particle is considered to be surrounded by a sphere of  influence of radius R, so that if the spheres of influences of two particles overlap the two particles will join and become a double particle. This double particle will then continue to describe Brownian motion but at a reduced rate, due to its increased size, until such time as it encounters another single, or double particle. An overlap in the spheres of influence of the two particles will then result in the creation of a triple, or quadruple particle. This process will continue until all the particles have coagulated to form one single aggregate. The problem that we wish to solve is the specification of the concentrations v  u  \ ^2i z  ' • • °f single, double, triple,(etcetera) aggregates at time t—given that at  time t = 0 there were VQ single particles. Before tackling this problem, however, it is illuminating to consider the following more elementary problem. What is the rate at which surrounding Brownian particles are deposited on the surface of a particle fixed at the origin and of radius R? The dynamics of the particles are governed by the diffusion equation, the solution of which represents the number of Brownian particles per unit volume at position r and time t, given some initial distribution of an ensemble of particles. The problem can therefore be formulated as the solution of the diffusion equation, with the initial condition of a uniform distribution of partcles for |r| > R, and the boundary condition that at |r| = R for time t > 0 we have a perfect absorber. Thus 2  [du/dt) = D V w ,  (2  67)  (2  68)  (2  69)  with boundary conditions  a; = v — constant, at w = 0 at  t= 0  |r| = R for t > 0,  for |r| > R.  CHAPTER  2: THEORY  23  where lo is the concentration of particles at position r and time t. As the problem has spherical symmetry the appropriate form of the diffusion equation is  (d/dt) {ru) = D {d /dr ) (rw) 2  2  (2 - 70)  The solution of this problem is  R 2R - + — f r ryj-n J  LO — U  ri'-*)W)  1/2 2  exp(-x ) dx  (2-71)  0  From the above it follows that the rate at which the particles arrive at the surface is given by tto  4nD[r*—)  r=R  = 4nDRu[l  + {xDt)  (2 - 72)  1/2^  Accordingly, Eq. 2-72 gives the rate at which particles describing Brownian motion will arrive at the surface of the sphere of influence of radius R of a particle situated at the origin. If, however, this particle is not stationary, but also describes Brownian motion, what would be the corresponding generalization of Eq. 2-72? It is clear that we must deal with the relative displacements between the two particles, and it can be readily shown  18  that the relative displacements between two particles describing  Brownian motion independently of each other and with the diffusion coefficients D\ and Z?2 also follow the laws of Brownian motion with diffusion coefficient £ )  1 2  =  D\ + Di- The probability that the relative displacements of two particles, initially together at t — 0, lies between r and r + dr is oo  W M r i j W j f n + rJdr.  W  /  (2-73)  -oo  2  = (A n*\z/2U n,i3/» / exp(-|r | /4i? 0exp(-|r (47rPif) / (47rJ3 0 ' J-oo 1  3  z  2  1  2  1  +r\ /4D t)dr l  l  CHAPTER 2: THEORY  or W  {  T  )  ( R 5 ^ W ^  ~  e  x  p  H  r  |  V  4  (  i  ,  ,  +  Z ) ! ) , )  (  -  2  "  7  4  )  Consequently, we conclude that the relative displacements do follow the laws of Brownian motion with the diffusion coefficient (D\ + D ). The subsequent required 2  generalization of Eq. 2-72 is  4,(  g |  +  i > ) R , ( , i  R +  |  M  i  >  i  +  g  ;  )  |  |  1  /  ( -75)  , ) .  2  We now generalize this further and consider two sorts of particles with concentrations U{ and v , with respective diffusion coefficients £>, and D . Let R k  k  denote  ik  the seperation distance within which two particles join to form a multiple particle. The rate of formation of the^multiple particles by the coagulation of the particles of the kind considered is then given by  J,  +k  dt = AnD^RiWt  + { ** y ) w  t  /t  dt  2  >  ( -  76  )  where D  ik  = Di + D . k  From here on in we shall ignore the second term in the parenthesis on the right hand side of Eq. 2-76, which implies that we restrict ourselves to time intervals 2  2  3  Af » (R /D). In most cases of practical interest this is justifiable as R jD ts 10~ 4  10~ second. With this understanding we can write  J dt ~ 4nD R w dt. l+k  ik  ik  k  (2 - 77)  Using this we can now write down the fundamental differential equations which govern the variations in V\,v ... ,v ... (of single, double, . . . k-fold . . . particles) 2  k  CHAPTER  2: THEORY  with time. So, considering the variation of the number of fc-fold aggregates with time, we have in analogy with the equations of chemical kinetics,  In this equation the first summation on the right hand side represents the increase in v due to the formation of fc-fold particles by the joining of an t-fold and aj-fold k  particle (with t + / = fc). While the second summation represents the decrease in v  k  due to the formation of (fc -I- j)-fold particles in which one of the interacting  particles is k-fold. A general solution of this equation is not feasible but a special case considered by Smoluchowski appears sufficiently illustrative of the general solution. We first assume that Ri = |(-Rj-f R ). Secondly, we note that the diffusion k  k  coefficient is inversely proportional to the radius of the particle. And since, on the basis of experimental evidence, it appears that the radii of the spheres of influence of various multiple particles are proportional to the radii of the respective particles, we can therefore make the additional assumption  DiRi = DR  1,2,...).  (i=  (2 - 79)  This leads to  DikRik = \(Di + D )[Ri  + R)  k  = ±DR{R7 l  l  = -DR(R, +  k  l  + R - )[R k  + R)  i  2  k  l  1  R ) R- R; . k  (2 - 80)  25  CHAPTER 2: THEORY 2 6  Finally, for the sake of mathematical simplicity we make—although not very plausible—the assumption that 72, = R . With all these assumptions k  DR lk  = 2DR,  lk  (2-81)  and Eq. 2-78 becomes  1 dv = *"DR(dt 2  °°  k  v  ~ vtY, i)  v  (*=1,2,...).  (2-82)  If we now let T = 4nDRt,  (2-83)  Eq. 2-82 takes the more convenient form of 0 0  du  -^7=  E  ^  -  2  ^  E  ^  (* =  1,2,.--).  (2-84)  This leads to d  '°°  oo " ( E ^ ) fc=i  or,  2  (2-85)  CHAPTER 2: THEORY  Thus using Eq. 2-84 we can successively obtain solutions for u\, vi, etc. Considering the equation for V\ we have  dT  = -2./, £ > * = ^  1  (1 +  0  ,  u t)  2 - 87  0  or in other words  again using the boundary condition that V\ = uq at t — 0. Proceeding in this manner we can prove by induction that  v  k  = ^\M  k  l  k+l  ~ l { \ + v T) \. Q  14  Extensive comparison with experiment '  15  (2 - 89)  has lead to the general conclusion  that Smoluchowski's theory gives a fairly satisfactory account of the broad features of the coagulation phenomenon. With this result we can predict that the average size of aggregate in the solution, k, will grow with time in a manner governed by  £  *W =  kv  k  ^ o — •  (2 " 90)  fc=l  This will be reflected in an increase in the average size of scattering particles obtained by measuring the average diffusion coefficient of the system. However, since this average diffusion coefficient—which can be obtained from the first moment of the distribution of decay rates of the correlation function G r ^ ( f ) — i s an average in which each contribution to the sum is weighted by the relative intensity of the  27  CHAPTER 2: THEORY  scattered light from that contribution, (2-91)  2  D = t/q  ( 2  • W -  -  9 2 )  For solutions of particles small compared to q~* with equal polarizability per unit mass  (l ) oc v M\ oc J/**;  (2 - 93)  2  k  k  2  E t  h  u  s  D  =  ±  Duk k  k  z Z ^ k  '  (  2  "  9  4  )  3  Efc ^ a n d  *"  =  £ ^  ( 2 _ 9 5 )  '  k  Consequently, by substitution of Eq. 2-89 in Eq. 2-95 we see that  Kz =  ,  r .  .  (2 - 96)  This summation can be evaluated in the following manner. If we make the substitution a —  I/QT/{\  + VQT),  E fc  K 1  k u {u T) ' 3  0  (1  +  0  I /  0  T ) * + »  2  t/pc  =  (1  +  , 2  2  5  _  3  3  3  (l + v )*da oT  3i^oa +  fc  A  ^) Y  VQO.  3  Y  d  2  * fc  (1 + ^ ) » ? Y  B  CHAPTER ,  U }  d  00  2  (1 + v r) da  3  THEORY  k  I  3  1 \  \l-a)  0  3t/ g  a  0  +  d  "Q  2:  2  /  1 \  (l + i / r ) * 3 a » V l - a) 0  2  (1 + I/O'")  Vl - a/  2  =i/o!6(i/o0 + 6(i/ 0+!]• 0  (2-97)  Similarly, X > * * = [l + 2(i/ 01fc 2  (2-98)  0  Therefore IS^QT) + 6(^ r) + 1) 2  0  ( 2 _ 9 9 )  [1+2(^)1  This relationship, though, does not completely describe the virus that is of interest (CRSV), as it assumes that whenever two viruses or aggregates of viruses come to within a distance R of each other they always join to form a multiple particle.  However this is not always observed to be true. Previous studies by  Tremaine et. a / .  8,4,5  indicate that there is a temperature dependent probability  that upon collision a multiple particle will be formed. We therefore hypothesize that this temperature dependent probability is of the form  p(t)  where E  agg  = { e x p i - W M r - r )]  Y//^.  e  (2 - ioo)  and T are termed the energy of aggregation (E.A.) and temperature of c  aggregation (T.A.). The rate of formation of multiple particles is then given by  J dt i+k  = i*D R ViV xk  ik  k  exp\-E /k [T agg  B  - T )}. c  (2 - 101)  CHAPTER  2:  THEORY  Following the same procedure as before we arrive at the conclusion that _ |6(,„,)'+ 6(, ,) !] 0  [1  _  +  2I/ T]  +  V  1  0  and T = 4nDRexp[-E /k {T agg  B  - T )]t. c  (2 - 103)  CHAPTER  S: EXPERIMENT  chapter 3 experiment  3.1 T h e Correlator The correlator used in these experiments was a Malvern K7023 digital correlator.  Its operation and the method of photocount autocorrelation have been  discussed in detail by many o t h e r s  1 0 , 1 1  1 6  ' '  1 7 , 1 8 , 1 9  . Here I shall assemble the impor-  tant concepts relevant to macromolecular solutions. Prior to processing by the correlator the signal is digitized.  A n example  of this process is illustrated in Fig. 3 - 1 , which shows a typical trace of light of fluctuating intensity. About three coherence times are spanned in this figure. At any instant the probability (per unit time) of detecting a photon is proportional to 20  the intensity . The second line shows a typical distribution of detected photons, or photocounts, per small time interval of length T . The bunching of photons due t  to fluctuating intensity is evident: when the intensity is high many photons are detected per interval, and when it is low few are detected.  The third line shows  the number of photons detected in the sample time (S.T.) T , centered at time 3  defined to be the number of phototcounts, «(£,). The photomultiplier, discriminator and amplifier assembly used was an E . M . I D307K which produced pulses of height - 1 . 2 V and width 30ns, with a uniform rise and fall time; each pulse corresponding to a single photon.  The method of  integrating the total signal arriving between two sample time clock pulses has the  31  CHAPTER  S:  EXPERIMENT  advantage that no photocounts are lost due to dead time between S.T.'s, therefore no information is lost.  •z  m  — or  <  PHOTONS  F i g u r e 3-1 Typical trace of fluctuating light intensity, and typical distribution of detected photons (photocounts).  The undipped photon number, n, and clipped  photon number, n , for clipping levels 0-4, are also shown. k  The operation of the correlator is illustrated in Fig. 3-2. The object of operation is to construct the sum N  Cm = E ('i)*('i-in) i n  where  l^+i -  n  = T , t  CHAPTER  S:  EXPERIMENT  and M is the number of correlator channels. The correlator uses a shift register to store the delayed countsrc,t(<,_ )in a single bit quantized form. Those that arrive m  during the S.T., T , centered at,  are counted in the scaler. Thus for the duration  S  of this sample time the scaler will store the value n(f,).  ™ . , .  T , r »  COUNTER INPUT » —  J  CLIPPING GATE  n(0)  n(0)  SHIFT  n(0)  REGISTER  n (T.)  n(3V  k  _i AND GATE  AND GATE  AND GATE  AND GATE  u 2  n(0)n(0) k  n(0)n(T) k »  CORRELATOR  n(0)n(2T) k  •  n(0)n(3T)  STORE  Figure 3-2 Schematic operation of a digital clipped correlator.  If the pre-set clip level, k, is exceeded a logical 1 is stored in the first element of the shift register. Otherwise, the input is zero. At the next S.T., at f,+i, the scaler counts the new number of photocounts n(t j). Within this sample time the l+  contents of the shift register are shifted one place to the right, and the first element of the shift register is set equal to the clipped value of n(r; ). Thus at time ti, the +J  contents of the r element of the shift register are represented by n^(<, - TT). AS the th  register counts only l's and O's multiplication can be performed by using the register to control 'and gates' that lead to separate storage channels. The instantaneous  CHAPTER S: EXPERIMENT  signal pulses, n(£,), are thus applied to each 'and gate' and the resulting product th  n(£,)-rifc(f, - rT ) is stored in the r a  storage channel. By repeating the process for N  samples where N is a large number, the contents of the store form the average of the 2  product G ' ' ( r T ) = TV (n(t)nk(t - rT )). 5  a  The progress of the correlation function  is monitored on an oscilloscope screen, and data are collected until a correlation function of sufficient accuracy is achieved.  Thus the correlator store contains M  correlation coefficients for delays between T and MT . For the Malvern K7023, M s  S  is 24, and the S.T.(!T,) can be varied from 1 s to 50 ns. There is facility for two input signals A and B; and in addition to the 24 A  B  channels the total number of counts in A and B , (n (t)) and (n (t))., are monitored, as are the number of clipped counts (nv(t)) and the total number of sample times N. The aforementioned four quantities are stored in channels 0 to 3 of the correlator output and the correlation function occupies channel 4 to 27. When the correlator is being used in single clip mode, input B is unused and correspondingly channel 3 will always contain zero. A detailed description of the controls is contained in appendix 2.  3.2 D i g i t a l C o r r e l a t o r C o m p u t e r Interface To extract the data from the correlator in a form that would facilitate ease of manipulation, a microcomputer (built by the U.B.C. electronics shop) was used. This interfaces the output of the correlator to the U . B . C . Amdahl 5850 mainframe computer via a control terminal.  This was done in such a way as to allow the  terminal to be used in its normal fashion when the correlator was not being used! A block diagram of the interface is shown in Fig. 3-3. The hardware consists of 4 circuit boards, the C R T terminal, a Cromemco single card computer (based on the Z80 microprocessor), an Industrial Micro systems 8k static R A M , a Cromemco T U - A R T serial communication board, a U B C  CHAPTER S:  EXPERIMENT  2 8 0  MICROCOMPUTER 8K ROM  TERMINAL  8K R A M S E R I A L I/O. P O R T I  NORMAL  MODE  1  S E R I A L I/O. P O R T 2  LINE S E R I A L l./O. P O R T 3 P A R A L L E L INTERFACE  DRIVER SEND MODE  GET MODE  AMDAHL M A L V E R N DIGITAL CORRELATOR  5850  Figure 3-3 Digitalcorrelator interface block diagram.  physics wire-wrap parallel interface, a Soroc IQ-120 terminal, and a UBC physics chassis and power supply. The Cromemco single card computer has 8k of programmable ROM which contains the interfacing program and Z80 Assembler routines to control the interface. This board also contains a serial communications device (UART) to allow communications between the terminal and the microcomputer. The 8k static RAM is used to store the data from the digital correlator prior to sending it to the Amdahl 5850. The Cromemco T U - A R T is used as an RS-232 communications device to allow communications between the microcomputer and the Amdahl 5850  CHAPTER  S: EXPERIMENT  The Parallel Interface board was constructed by the Electronics shop to interface the Malvern Digital Correlator's calculator output to the Z80 Microcomputer. It consists of 42 data and status signals and 1 output command signal to correlator. The communication line to the Amdahl 5850 is a leased line from the computing center running under Hardcopy support. Transmission speed to the Amdal 5850 is at 300 Baud while reception is at 9600 Baud. This means that while the terminal is used in normal mode the communications appear to take place at the typing speed of the operator. The transmission speed is only apparent when sending a block of data.  3.3 The Index Matched Cell and Temperature Control Device The sample cell for the experiment was a I cm bore pyrex test tube that 3  held approximately 3 c m of*solution. The cell was mounted in a Malvern Systems Index Matched Cell (I.M.C.). This is illustrated in Fig. 3-4. Ease of optical alignment is facilitated by a refractive index matching fluid — water in this case, that surrounds the specimen cell, filling the internal capacity of the I.M.C. For temperature control the central block of the I.M.C. contained a heating coil that was connected to a Malvern temperature control unit RR56. With this unit the temperature of the specimen cell could be set to an accuracy of •05% absolute with a temperature stability of 0.05 C°. Prior to the experiment the Malvern temperature control unit was calibrated against a thermistor embedded in a small copper block and subsequently inserted in the I.M.C. in place of the sample cell. The thermistor was then used as one arm of a Wheatstone bridge and balanced against a decade box using a Hewlett-Packard 419A D . C . nullmeter. The thermistor was calibrated against a Hewlett-Packard 2804A quartz thermometer, which had previously been calibrated by comparison with a J arret t triple point of water cell.  36  CHAPTER SPECIMEN C E L L  INSULATING  CONTAINER  SPECIMEN  S:  CELL  TEMPERATURE  CONTROLLED  HIGH  CAPACITY  HEAT  EXPERIMENT  THERMAL SINK  F i g u r e 3-4 Index Matched Cell.  3.4 O p t i c s A diagram of the apparatus is illustrated in Fig. 3-5.  The laser source  was a Hughes 10 mW helium-neon laser. The first two polaroids that the beam encounters are mounted in such a way as to allow independent rotation. The axis of polarization of the second defines the incident polarization of the beam (u,) while the angle between the axes of polarization of the two defines the incident intensity. The beam is focused, by a microscope objective, onto an optical filter (consisting of a 10 um pinhole) which is at the focal point of a 50 mm lens, which collimates the emergent beam. A 400 mm lens focuses the beam to a diameter of w  1  l/io"  mm  at the scattering volume. The polaroid in front of the P . M . T . defines the polarization of the scattered light collected: i i / . The entrance aperture of the P . M . T . was variable from 1/2 mm  CHAPTER  LASER  3  LENS NUMBER I 50mm  LENS NUMBER 400mm  I) 2 POLAROIDS  2  8:  EXPERIMENT  SPECIMEN C E L L  POLAROID  2) MICROSCOPE OBJECTIVE 3) OPTICAL F I L T E R  IO>m PHOTOMULTIPLIER WITH BUILT IN LENS  TO CORRELATOR  Figure 3-5 A schematic diagram of the apparatus.  to 6 mm in diameter. For all experiments it was set to 1/2 mm for maximum angular resolution. Two screws in the photomultiplier tube (P.M.T.) position the P.M.T.'s lens so as to form an image of the scattering volume on the pinhole aperture of the P.M.T. To ensure that this was done accurately a long necked pipette of inside diameter « 1mm wasfilledwith spheres and placed inside the I.M.C. and aligned by two locating holes. The intersection of the beam with the pipette provided a bright target for alignment of the P.M.T. lens. The two screws in the P.M.T. were adjusted to achieve a maximum count rate observable on the correlator's rate meter with this target. To ensure that the scattering volume was at the focal point of the P.M.T. lens the P.M.T. was swung around to the zero degree position, and the pinhole aperture of the P.M.T. was illuminated by a desk lamp shone through an observation window in the P.M.T. The P.M.T. was then slid back and forth on its  CHAPTER  S:  EXPERIMENT  rail until a sharp image of the pinhole could be seen through a telescope placed between lens number 1 and lens number 2 and focussed at infinity.  3.5 Preparation of Sample  The Spheres The sample cells used were Corning 10 x 75 mm Pyrex test tubes. The Latex Spheres used were manufactured by the Dow Chemical Company and marketed by Seragen diagnostics. They were made of styrene butadiene, and were of diameter 0.087 ^m, with a standard deviation of 4-6 nm. The samples were prepared as follows. The concentrated stock suspension of 10% solids was shaken and allowed to settle for a few hours. Then 0.05cc of the suspension was withdrawn by pipette from just below the surface, in the hope that any impurities or coagulated spheres would have settled out of this region faster than the particles of interest. This drop of suspension was added to a sample cell which contained distilled deionized water which had been passed through a 2 um Millipore filter. This cell was then centrifuged for 20 minutes to remove further impurities or coagulated spheres, and decanted into a second sample cell. This sample was then used as a base solution to be diluted down to the required concentration for a particular run.  The Viruses The sample of CRVS used was provided by the Agriculture Canada Station, Vancouver, Canada. The sample was prepared in 0 1 M sodium acetate buffer (pH 50). The acidity of the buffer was necessary to avoid the virus swelling. To avoid the accidental destruction of the virus sample, which was in short supply, the sample was split into four aliquots, and placed into four separate sample cells. Glass beads were added to the cells to raise the level of the solution to the level of the laser  CHAPTER  beam.  S:  EXPERIMENT  The concentration of the sample was measured, by Agriculture Canada, 2  4  spectrophotometrically using the extinction coefficient of 6.5cm /mg at 260 n m , after each experiment.  C l e a n i n g the Cells The cleaning procedure for the cells was as follows. (i) The cells and all pipettes, syringes, and any other equipment that was to come into contact with the sample were immersed in a solution of equal parts of alcohol and a 30% concentrated hydrochloric acid solution, and thereafter left for 4 to 6 hours. (ii) The equipment was then thoroughly rinsed with distilled water until no trace of acid remained. (iii) The equipment was theTi flushed with acetone squeezed out of a wash bottle. (iv) Further washing was subsequently done with acetone that had been freshly distilled, using it as it fell from the still into a beaker. (v) Drying was done in a clean oven for 20 min at a temp of 75 °C.  3 . 6 T h e Experimental Procedure Each repetition of one of the following experimental procedures constitutes a run. Two experimental procedures were followed. Which one was chosen depended on whether the sample was in a dynamic state or an equilibrium  state. An Equilib-  rium state is one in which the correlation function is not changing with time—Latex spheres is an example. A dynamic state is one in which the correlation function is changing with time. A n example of this could be a system that was undergoing a chemical reaction, or in this case a system that is aggregating.  A n Equilibrium R u n An equalibrium run is a process by which correlation data are taken on a system that is in an equilibrium state. Prior to the run the sample was placed in  CHAPTER S: EXPERIMENT 41 the I.M.C. which had previously been raised to the desired temperature for the run. The desired sample time, clip rate, and total number of sample times to be taken were set using the correlator controls. For most of the runs the number of photons per sample time (r) was in the range 0-1 to 1 and the correlation functions were of the form exp —Tt, with a small correction term for the polydispersity. The minimum error was obtained when the sample time T was chosen so that MTT, ~ 2 — 3 .  21  t  It should be noted that when r lies in the range 10-100, T should be chosen so s  that MTT  t  ~ 1 — 2. The clip rate was chosen by performing a trial run before the  experimental run and adjusting the clip level so that the second channel accumulates 22  at no more than half the rate of the fourth channel . For the vast majority of sphere runs the clip rate was zero due to the low count rate. As has been previously mentioned the total number of sample times was chosen, from experience, to ensure a high signal to noise ratio. Prior to each experimental run a computer file was created that subsequently contained the data from the run. The creation of the file was facilitated by the program C O R R F I L E C R E which created a file and named it with the date and time that it was created, eg JAN21113145. The command $RUN C O R R F I L E C R E is coded to be activated by the C T R L - F key. Information about the particular run is then written on the first line of the file : date, scattering angle, sample time, clip rate, temperature, and multiplication factors for the first four channels. The run is then initiated by clearing the correlator store of all previous runs and pushing the start button. The correlator will stop automatically after N sample times. The P U N C H readout mode is then selected on the correlator and the data are then loaded into the computer file via the Z80 by depressing the E S C - G the ESC-S keys at the terminal. The run is now complete and data are ready to be processed.  Non-Equilibrium or Dynamic Runs A dynamic run is essentially a succession of short equilibrium runs on a  CHAPTER  S:  EXPERIMENT  single sample that is in a dynamic state: aggregating, in this case. Such a run gives information on the progress of the dynamic process: the aggregation. More specifically, a measure of the mean scattering particle diameter ( f ) is a measure of the mean number of viruses in an aggregate measured by sampling k  tx  (fc).*  The rate of aggregation can be  at regular intervals of time. Thus each 30 min dynamic  run consisted firstly, of a two minute period to allow the sample cell to reach the temperature of the l . M . C . Correlation data was then taken over ten one minute periods, each being followed by a two minute period during which the data that had just been collected would be processed, and, depending on the results, certain adjustments would be made to the correlator settings in preparation for the next one minute of data collection. Consequently, statistical accuracy could be maintained despite the changing intensity of the scattered light and increase in particle size. TV  Each dynamic run therefore makes a measurement of the aggregation rate that corresponds to the two experimental variables, temperature and concentration. By preparing samples of various concentrations and performing a series of dynamic runs at different temperatures the dependence of the aggregation rate on temperature and concentration could thus be investigated.  * As explained in Sec. 4.2  CHAPTER  4: DATA ANALYSIS  CHAPTER 4 DATA ANALYSIS  4.1 Mono dispersive Data Analysis The desired system parameter—the diffusion coefficient—can be obtained by fitting the data obtained from the correlator to the expected analytical form of the correlation function, and extracting the material parameters from the fit parameters. For equilibrium runs the basic equation for data analysis is  (C ) m  = N (n) (n ) [1 + Texp (-2rmr.)],  (4-1)  k  (1 + *)  o  /  2  and  T =Dq .  (4 - 3)  E q . 4-1 relates the average correlation store contents, ( C ) , and the average single m  channel counting rates, (n) and (n ) , to the decay rate T. However, due to the k  random nature of both the intensity fluctuations and the photodetection process one obtains only an approximation of the above quantities.  43  CHAPTER  4: DATA  ANALYSIS  A set of experimental signals can be defined by  Q m  m  'N(n)  (n )„  ( 4 - 4 )  - i,  k  t z  N vv  ( 4 - 5 )  N, N 1=1  ( 4 - 6 )  N  which must be fit to the function 7 exp( — 2TmT ) in order to obtain an experimens  tal value for T. In practice it is more convenient to fit In S  m  (In Tf - 2YmT )  than to fit S  m  t  to the straight line  directly to an exponential. The data analysis thus  consists of a two parameter minimization of the sum  N  On S  m  - In 7 + 2rmT ) , 2  e  Xm  ( 4 - 7 )  m = l  where the X ' ' m  s  a  r  e  weight factors. A reasonable fit is obtained by taking the weight  X m to be inversely proportional to the variance in the measured quantity 1 X  m  m  2  2  " ((6lnS ) )  ~ ({6sZ)*y  is the expected variance in S .  tively weak m dependence  m  20  , i.e.  S  m  where ((6S ))  2 0  ( 4  ~  8 )  One can, however, neglect the rela-  of ( ( 6 S ) ) and use 2  m  Xm  oc S  2  .  m  (4  - 9)  CHAPTER 4- DATA ANALYSIS  4.2 Polydispersive D a t a Analysis Polydispersive data analysis consists of fitting to the function  S = l n 7 - 2 f TnT", + fi {mT )\ m  2  (4 - 10)  s  with the same weighting as before. A system of aggregated particles will contain many different size particles. The Einstein relation therefore generalizes to  knT 7777T'  Dk=~  U-ll)  where (///o) is a shape factor, and r\ is the viscosity of the solvent. In this case the solvent was sodium acetate, which has approximately the same viscosity as water, 28  given as a function of temperature by the relation ,  TI =  s;  where  »?20  1-3272(20 - T) - 0.001053(7" - 20) —  —  r  +  .  o  s  —  2  -  (- -  1 2  >  is the viscosity of water at 20°C and 7* is measured in ° C . It is assumed  that the viruses aggregate along no preferred axis and therefore form approximately spherical aggregates. This being so, then (///o) — 1 - It is also possible to calculate an experimental estimate of the average number of particles in an aggregate {k ). ex  1  As the diameter of a single virus has been found to be 34nm then  * „ = ( ( f / r f ) ~l)Pf+ 8  1,  (4 - 13)  where d is the diameter of a single virus, Pj is the packing fraction, and f is the mean diameter of the aggregates in the system.  It was assumed on consultation  with Tremaine and Ronald that the aggregates formed some sort of close packing structure, and so the packing fraction was estimated to be 0-7.  CHAPTER  4: DATA  ANALYSIS  The program C O R R . A G was used to fit the data, and is listed in appendix 3. From the parameters of the fit, values of D, Q, and k  tx  were calculated, using  Eq.2-64,65,66, and Eq. 4-13. Fig. 4-1 through 4-4 give an example of the output of C O R R . A G . By thus using the program C O R R . A G to analyse the data produced by each dynamical run it is possible to measure the rate of increase of k  cx  with time.  It is expected that this will be governed by Eq. 2-99. If this is rewritten as [6(Jtt)» + 6 ( M ) + I) k  t  x  [  t  )  (l-r2(*0)  -  (  '  1  where  R = *o(r/0,  (4 - 15)  then as R is a constant for &ach particular concentration and temperature, it can be thought of as the rate of aggregation. The next stage of the analysis consists of fitting the values of k  cx  obtained from the correlator runs to Eq. 4-14 to obtain  the dependence of the rate constant on temperature and initial concentration. The program ANALYSIS was used for this. ANALYSIS performed a two parameter fit to the function,  F  . .  2  I6i + 6 i + 1]  \T+lx\ with  Pi and P where t  a  2  16  <"- >  x=Pi{t + P ).  (4-17)  2  are the two parameters in the fit, representing R, and t  a  respectively,  is the time that has elapsed between the system commencing aggregation  and the experiment starting *. The timing of the runs commenced two minutes after placing the scattering cell inside the I.M.C. This is because it was calculated that the contents of the scattering cell would take two minutes to reach the temperature of the I.M.C. This conclusion was arrived at by solving the heat equation for a * see below  CHAPTER  4: DATA  ANALYSIS  cylindrically symmetric tube 1cm in diameter which is initially at temperature T — TQ with wall held at a constant temperature T = T u. The series of Bessel functions ce  that results is plotted in Fig. 4-5. This, however, is the maximum time. If there is convection in the cell the contents will reach the temperature of the I.M.C. faster. This approach does have the limitation that some aggregation may occur before t — 0. Indeed, experiments carried out a good deal below the temperature at which aggregation commences indicate that there is some residual aggregation left from some previous passage of the system past the temperature of agrregation. To allow for this eventuality we therefore performed a two parameter fit on the data. The parameter P  2  takes into account any aggregation in the sample at t = 0 in the  following manner. It is assumed that all aggregation present at t = 0 had taken place at the rate R from time t = ( —P ). ln other words, we are assuming that 2  at the time t = - P , the initially monodispersive system was instantly raised to 2  the temperature of the I.M.C. The system then proceeded to aggregate at the rate R until t = 0, when the experiment is commenced with the system in a partially aggregated state. It was also noted that for large values of k , tI  this equation was l  often not a good description of the data. This was because as f approaches q~ , Eq.2-93 is not true and  is no longer proportional to v k?. Thus k  E D  *  ±  D v k? k  t  T ^ r -  (  4  -  1  8  )  k  It was further noted that once the system had reached this stage of aggregation, it becomes somewhat turbid. We shall therefore herein refer to data collected while the system was in this state as data collected while in the turbid regime. Data taken in this regime gives an indication as to whether the system is still aggregating, but values of k  tx  so obtained have no precise correspondence to k. The program  ANALYSIS can thus perform a fit to any particular range of t values to ensure a  4: DATA  CHAPTER  ANALYSIS  good fit. The relationship between the aggregation rate R, and VQ and T is governed by R = ex \-E /k (T P  agg  B  - T )). (^p) c  •  (4 - 19)  If this is rearranged it becomes  The program L E A S T S Q U A R E then performs a least squares fit to the function,  =PiT  +P, 2  for the values of R and T 'corresponding to each concentration tried. E  agg  (4-21)  Values of  — Eaggi^o) and T — T (yo) can then be extracted from the two parameters, c  c  and the functional form of their dependence on VQ can be investigated.  I  CHAPTER 4: DATA ANALYSIS  49  FILE = sep16.112430 THE EXPERIMENTAL PARAMETERS RUN#=16098501 THETA= 90.000 CLIPRATE= 0 TEMP= 30.00  STIME= 28.50 MULT.FACTS.= 0 0 0 0  THE FIRST 4 CORRELATOR CHANNELS ARE NCOUNT= 1256642 NCLIP= 1131904 ZERO= 0 NSTIME= 10000918 XN BL CF B  = = = =  NCOUNT/NSTIME = 0.1257 XN * NCLIP «= 0. 1422E+06 (XN+1)/(CLIPRATE+1) = 1 . 1 2 6 EXPERIMENTAL PARAMETER (DARK NOISE)  INPUT = DATA READ IN; X = MULTIPLE OF STIME; Y1 = G(T) = (INPUT - BL)/(YMAX - BL); Y2 = LN G(T) + LN(B/CF) = LN( INPUT/(BL - 1) ); WT = ( INPUT/(BL - 1 ) ) ** 2 . 0 • STAT WEIGHT INPUT 228898 219415 211147 203121 197531 191543 185916 181398 177679 172154 169597 167373 163745 162187 159012 157633 155836 154589 153990 152876 151029 150345 149426 148913  X 0. 0 28. 500 57. 000 85. 500 114. 000 142. 500 171 .000 199. 500 228. 000 256. 500 285. 000 313. 500 342. 000 370. 500 399. 000 427.500 456. 000 484. 500 513. 000 541 .500 570. 000 598. 500 627. 000 655. 500  Y1 OR G(T) 1 . 00000000 0. 89058661 0. 79519171 0. 70258898 0. 63809246 0. 56900382 0. 50408041 0. 45195240 0. 40904319 0. 34529662 0. 31579435 0. 29013419 0. 24827486 0. 23029894 0. 19366628 0. 17775559 0. 15702206 0. 14263439 0. 13572323 0. 12287009 0. 10155970 0. 09366781 0. 08306450 0. 07714558  Figure 4-1 CORR.AG output Pg.l  LN G(T) + LN(B/CF) -0 .4952995 -0 .6111743 -0 .7244727 -0 .8482835 -0 .9445734 -1 .0591698 -1 .1803198 -1 .2894773 -1 .3892355 -1 .5586510 -1 .6479654 -1 .7327156 - 1.8885193 -1 .9636812 -2 .1369247 -2 .2226458 -2 .3466730 -2 .4427805 -2 .4924459 -2 .5919371 -2 .7824173 -2 .8633137 -2 .9834442 -3 .0573654  WT 2 .5901680 2 .3799963 2 .2040091 2 .0396404 1 .9289207 1.8137436 1.7087460 \ .6267042 1.5606861 1.4651365 1.4219360 1.3848867 1.3254995 1.3003969 1.2499819 .22B3945 1 .2005463 1. 1814108 1.1722717 1.1553726 1.1276236 1.1174326 1.1038141 1.0962486  CHAPTER  4: DATA ANALYSIS  PLOT O F C O R R E L A T I O N F U N C T I O N  o  r—  O  LU  o o  i  0  r i i i i i r 100 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0  DELAY TIME m i c r o s e c  Figure 4 - 2 CORR.AG output Pg.2  50  CHAPTER 4: DATA ANALYSIS  51  FITTING POLYNOMIAL : Y «= P1 + P2*X + P3*Xa2. INTERMEDIATE ESTIMATES OF PARAMETERS, SUM OF SQUARES 0.0 0.0 0.0 115.77 -0.48848 -0.41372E-02 0.28945E-06 0.16555E-01 -0.48848 -0.41372E-02 0.28937E-06 0.16555E-01 FINAL ESTIMATES OF PARAMETERS -0.48848 -0.41372E-02 SUM OF SQUARES EU0.382567E+00 E2=0.107414E-01  0.28937E-06  0.16555E-01 0.304686E-02 0.855470E-04  0.475851E-05 0.133605E-06  ND«  K SCATTERING VECTOR ( / c m ) B  D=DIFFUSION COEFICIENT  (cm/sec)  RL«=RAYLEIGH LINEWIDTH (hz) M2=2nd MOMENT OF LINE WIDTH DISTRIBUTION (hz**2) Q=THE QUALITY FACTOR (dimensionless) DIA=THE AVERAGE DIAMETER OF THE PARTICLES ( n m ) KBAR=THE MEAN NUMBER OF PARTICLES PER AGGREGATE  K «= M2 « KBAR =  187320. 289373. 11.22  D « Q -  Figure 4-3 CORR.AG output Pg.3  0.589532E-07 0.676246E-01  RL DIA -  2068.60 94.4532  CHAPTER  4: DATA ANALYSIS  PLOT O F L O G O F CORRELATION FUNCTION -0.25  T  DELAY TIME m i c r s e c Figure 4-4 CORR.AG output Pg.4  52  CHAPTER 4- DATA ANALYSIS  T E M P E R A T U R E O N C E  35  O F  P L A C E D  S A M P L E IN  53  C E L L  I.M.C.  T —  •  5H i  O  i  200  i  400  1  600  r—  800  TIME s e c o n d s Figure 4-5 The average temperature of the cell after it has been placed in the I.M.C.  CHAPTER  5:  RESULTS  CHAPTER 5  RESULTS  5.1 Introduction This chapter is organized into four sections.  The first section will cover  experiments in which the scattering sample was polystyrene latex spheres. The diffusion coefficient is measured, and its dependence on temperature and scattering angle is discussed. For the second and third sections the scattering sample was CRSV. In the second section the runs were performed at a temperature below which the virus is expected to aggregate for that particular concentration. The diffusion coefficient is measured and its dependence on temperature and scattering angle is discussed, along with the possibility of residual aggregates being present from previous aggregations. In the third section the aggregation behaviour of CRSV is studied. The dependence of the aggregation rate upon virus concentration and temperature is discussed and the temperature of aggregation (T.A.) and aggregation energy (A.E.) are calculated for each concentration sample. In the fourth section a few suggestions for further investigations are made.  5.2 Latex Sphere Results 24  The latex sphere samples were purchased from Seragen Diagnostics , and were 0.087 /xm in diameter. They were suspended in deionized distilled water at a _5  concentration of 3-35 x 10 g/ml, which corresponds to a mean spacing between  CHAPTER  5:  RESULTS  particles of twenty-five times their diameter. Experiments were performed at scattering angles of 30° to 140° at 10° intervals. The entrance and exit apertures of the I.M.C. prevented readings outside this range. For each run f and / i were extracted 2  from the fit to the correlation data, and the corresponding values of D, f,and<5 were calculated. In Fig. 5-1 a graph of the Rayleigh linewidth, f, versus the square of the scattering vector q , is plotted, and a straight line is sketched through the data 2  points corresponding to the same temperature. As f = Dq , the increase in slope 2  of each line occurs as a consequence of the increase in D with temperature. To check that this is consistent with the Einstein relation for the diffusion coefficient the diameters corresponding to each run are plotted in Fig. 5-2, and in Table V-I the values of the mean diameter f, the standard deviation from f, and the the mean value of the quality factor Q are listed for each temperature. Table V - l 1  fa"" (nm)  Q  (•c)  f (nm)  26 30 34  98-4 100-7 99-7  3-7 3-2 2-8  003 002 002  temp  The scatter in the diameters in Fig. 5-2 showsfirstly,that any systematic deviation from the Einstein relation for the diffusion coefficient as the temperature is increased is less than the uncertainty in the measurement; and secondly, that the diameter seems to be independent of the scattering angle, within the accuracy of the experiment. The Q values were all ±30%. These non zero Q values suggest that the system was not entirely monodispersive. Possible reasons for non-zero Q values would be dust in the sample, or particles in the solvent which was onlyfilteredto 25  2 /jm. It has been also reported that samples of latex spheres of this size have a  CHAPTER  S:  RESULTS  MEAN PARTICLE DIAMETER vs SCATTERING VECTOR FOR THREE DIFFERENT TEMPERATURES 110-1  0  100,000  200,000  300,000  SCATTERING VECTOR meters ** - 1 Figure 5-1 Variation in T with q for latex spheres. 2  CHAPTER  57  5: RESULTS  RAYLEIGH LINEWIDTH vs SCATTERING VECTOR SQUARED FOR THREE DIFFERENT TEMPERATURES 8000 T  ,  'o  700CH  6000H N  o,  <D  X  5000H  t—  Q  4000-1  O  •  3000H  _J  2000H  ioooH  I  1  I  2  I  3  I  4  I  5  SCATTERING VECTOR SQUARED  Figure 5-2 Variation in c with q for latex spheres.  1  6  7 1  *10°  CHAPTER  5: RESULTS  tendency to form into larger aggregates over long periods of time. Both these eventualities would lead to a larger value of f than expected. An inspection of Table V-I also indicates that the mean diameter, f is w 13% larger than the values quoted by Seragen diagnostics, although the standard deviation is approximately the same as that quoted by Seragen: 4-6 nm. Possible reasons for this are that Seragen are incorrect in their measurement, or that the dust or particles in the solvent are leading to a larger average diameter of particle in the system, or that the hydrodynamic radius of the particles is larger than the dry radius of the particles that is measured by inspection of the spheres under an electron microscope (which is how they are measured by Seragen); or that they are somewhat aggregated as mentioned before, or—more likely a combination of all four reasons. If the large value of f was due to the aforementioned impurities in the solvent, one would have expected a larger standard deviation than was measured, but the actual measurement was smaller than that quoted by Seragen. The same can be also said if the spheres were aggregated. A second measurement of the spheres is therefore clearly required. Evidence for an increase in the hydrodynamic radii with increasing temperature has been 26  reported , but this increase in the mean hydrodynamic radius with temperature was not seen. Having examined the results of the measurements on Latex spheres, the next step is to discuss the error in the results. CORR.AG provides an estimate of the error in the three parameters  Ci  k  (I\/i2>  and  7)  of the data fit to the Eq. 2.64.  = (n)(n ) [l + 7 e x p ( - 2 f i + /i2»-)] . fe  (5-1)  For each parameter Pj, CORR.AG produces a value  Pi = Pi ± Et  (5-2)  58  CHAPTER  69  5: RESULTS  where, J2w 6 (N 2  -M) •v/&„.  k  fori =  1,2,3.  (5-3)  k=l  N is the number of data points (twenty eight), M is the number of parameters in th  the fit (three), w is the statistical weight given to the k k  point, 6 is the residual  in the least squares minimization,  *=  f(x ,Pi,P2,P )),  (y -  k  k  3  (5-4)  and ba is the diagonal element of the covariance matrix corresponding to the i  tft  parameter. The errors in the parameters Tf, T, and^2 were ± 1 % , ±3%, and ± 30% respectively. As f is given by k Tq* B  (5-5)  67TT/r  we can calculate the error in f to be  <7  2  A f / r is given by CORR.AG to be 0 03. The only significant contribution to A g / ?  2  comes from uncertainty in the measurement of the scattering angle 0. Allowing for alignment of photomultiplier tube (P.M.T.), and adjustment of the lens in the P.M.T that focuses the image of the scattering volume onto the pinhole aperture of the P.M.T., A0 could be as large as 1°. Consequently the error in q can be obtained by differentiating the square of Eq. 2-22 with respect to 9, giving  Ag  2  A0  1  57'tan(f)'  (5-7)  CHAPTER  where 9 is in degrees.  5:  RESULTS  For 9 = 9 0 ° , then, Aq/q ~ 0 0175. The temperature was  known to an accuracy of « 0-05%, thus the uncertainty in TJ is about 0.1%. Consequently, A r / r ss 3|%. As the standard deviation in the sample provided by Seragen is « 5^% and the scatter in our data, from Fig. 5-2 is « 6%, the results seem to be within experimental error.  5.3 C R S V Results - I - Nonaggregating R e g i m e Nonaggregating samples consisted of solutions of CRSV that were either at temperatures below the temperature at which aggregation commences, or at a temperature close enough to T . A . that aggregation took place very slowly, (slowly implies k  ex  changes by less than 50%, from 1 to 1-5, over a period of an hour —  aggregation much slower than this is too slow to measure).  Experiments were  performed from 30°C to 50°C in 5 C° intervals. At the higher temperatures some aggregation was seen, but it is left for the next section to examine the aggregation in detail. The concentration used was 0-87 mg/ml, and angles 9 chosen for collecting data were: 7.5° , 9 0 ° and 105°. For each run D was extracted from the correlation data and the Einstein relation was used to calculate f. The results are plotted in Fig. 5-3. In table V-II the mean and standard deviation diameters, along with the mean value of Q, are listed. The results were seen to be independent of scattering angle and therefore the results are displayed grouped by temperature only.  Table V - I I temp (°C) 30 35 40 45 50  1  f (nm)  fa"(nm)  Q  34-5 34-5 36-4 370  0-55 0-89  0016 002  0-43 1 25 1-96  003 003 010  40-3  CHAPTER  5: RESULTS  M E A N DIAMETER O F V I R U S v s T E M P  <D -4—  o E  o c D  c CH  < <  30  35  40  45  TEMP celsius Figure 5 - 3 Values of f in CRSV for different temperatures.  50  61  CHAPTER  5: RESULTS  The mean diameter of the results performed at 30°C is 1-5% higher than the expected diameter of CRSV and is within the calculated experimental error. Indeed, the range of scatter of the results from the mean is only ± 2 | % , which is also within experimental error. As the apparatus was not changed from the experiments on latex spheres and the buffer in which the virus was suspended was unfiltered, it seems likely that the results obtained in Sec. 5.2 were an accurate description of the size of the particles. If the results at 30°C are assumed to accurately represent the monodispersive state of the system, the diffusion coefficient of CRSV at 20°C(i?20ui) can be - 7  2  calculated to be 1-24 x 1 0 c m / s , which is 20% lower than the.previously pub1  lished result .  But if the present result is used—along with the sedimentation  coefficient—to calculate the molecular weight (M) of the virus, a result is obtained that is consistent with theoretical calculations which assume that the viruses consist 6  of 180 protein subunits of 38000 daltons and RNA-1 of 1-5 x 10 daltons, giving 6  M = 8-3 x 10 daltons. Using the Svedburg relation  RTS  .  20W  .  where <f> is the specific volume of the virus (0-693 ml/ff), p is the density of water (0-998 g/ml), and S Q 2  1  W  is the sedimentation coefficient  1 3  - 1  (133 x 1 0 ~ s ) , M is  6  calculated to be 8-4 x 10 daltons—using the above value of  DQ. 2  W  The results at higher temperatures suggest that some aggregation is taking place. This is reflected not only in the higher mean values of particle diameter, but also in the increase in the standard deviation in f and the higher values of Q. A series of runs of one minute duration were performed on a sample at 50°C at 20 minute intervals.  The results are plotted in Fig. 5-4.  A n increase in the  mean diameter can be noted. The large scatter in the data are due to the short  CHAPTER  5:  RESULTS  run time of 1 min, and the increase in f is small as the virus is aggregating relatively slowly. The Q values, averaged over three runs, for each consecutive hour were: 0 02, 0 03, and 0-05. This increase reflects the increase in polydispersity in the system during the aggregation processes.  5.4 CRSV Results -II- Aggregating Regime In this section of the experiment the dependence on temperature and virus concentration of the aggregation rate (A.R.) of C R S V is determined.  This is  achieved by first measuring the A . R . as a function of these variables and then fitting the results to the previously hypothesized form of the aggregation rate. In doing so the dependence of the two aggregation parameters T . A . and E . A . on the experimental variables is ascertained. Experimental data was gathered by taking four samples of differing concentrations and measuring the aggregation rate for temperatures at approximately 1 ° C intervals through the range of temperatures over which this was possible. Experimental runs were limited to 30 minutes to avoid dammage to the virus. The temperature range was limited by two factors. First, its lower limit was determined by the fact that if k  cx  did not increase by more than the average uncertainty in  any one measurement during the 30 minutes of the experiment, the aggregation rate was deemed to slow to measure.  Second, its upper limit was restricted by  rate of aggregation. If the aggregation resulted in the formation of an average size of aggregate comparable to q~  l  before the first measurement could be made, the  aggregation rate was deemed to be too fast to measure. This is because the system was in the turbid regime where Eq. 2-93 to 95 are no longer valid. It was noted that as the system aggregated the amount of scattered light increased, as can be seen by reinspecting Eq. 2-93. Although the number of scattering centers decreases with time for an aggregating system, the number of particles in  CHAPTER  5: RESULTS  M E A N DIAMETER O F V I R U S v s TIME 46-i  X  X  in CD  "o E o c  H  *  X  D  g  X  X  X X  42-  r—  LJ  < <  bJ  X 40-  X 38-  1  20  1  40  1  60  1  1  1  1  1  8 0 100 120 140 160  TIME minutes  Figure 5-4 Increase in f with time as CRSV aggregates.  1  180  64  CHAPTER  5:  RESULTS  each aggregate [k) increases, and as this appears as the second power the total intensity of scattered light will increase. If the amount of scattered light is increasing then the average number of photons per sample time (n) recorded by the correlator will increase, so between each one minute run the mean number of clipped counts per sample time must be checked. If it is much greater than 0-5 then the clip rate must be adjusted accordingly. As the system aggregates f will increase and f will decrease since f oc D and D oc f . Therefore, as statistical error is minimized when MY T ~ 2—ZT s  be adjusted accordingly.* MTT  a  - 1  must  is reflected in the fourth column of page one of the  a  output of CORR.AG. ** To achieve maximum accuracy, ln G(24T )-ln a  G{0) ~ 2-3.  Column four must therefore span a range of 2 — 3. This was checked after each 1 minute run and T adjusted accordingly. s  Each 30 minute run produced 10 values of k  at 180 sec. time intervals.  tx  These points were used by ANALYSIS as a data set to fit to the function  [t]  _HRt?  k t x [ t )  + 6(JK) + 1]  [1 + 2(J20]  ~  {  '  An example of a bad fit to this function is shown in Fig. 5-5. The last three points correspond to mean aggregates much larger than q~ , where the system is in the x  turbid regime, and therefore Eq. 2-93 does not apply. Accordingly ANALYSIS is rerun excluding these points from its data fit. Fig. 5-6 shows the new fit. Here the fit is much better, but now on this expanded scale it can be seen that only the first three points fall on a straight line. Consequently ANALYSIS is run again with just those points, see Fig. 5-7. Now it can be seen that there is a good fit. The maximum point on this graph (k  ex  ss 25) corresponds to a mean aggregate size of  ss 0-1 ftm, which is approximately the same magnitude as q~ ss 0.06^m. The three l  * as explained in Sec. 3.6 ** see Fig. 4.1  CHAPTER  values of R obtained from the fits were: 41, 2-7, and 1-7 x 1 0  - 2  s  _1  5: RESULTS  respectively.  For the majority of experimental runs it was found that all ten data points could be used. The previous example is that of a system nearly at the maximum aggregation rate one could measure. Having obtained the aggregation rates in this manner, the next stage of the analysis was to fit them to Eq. 4-19  R = exp{-E/k (T-T ))-(^^y B  (5-10)  c  m  In Fig. 5-8 we therefore plot l / ( ^ f c ^ r )  v e r s u s  T for the four concentrations. The  data fit was done by the program LEASTSQUARE which performed a two parameter fit to Eq. 4-20. The four values of the energy of aggregation and temperature of aggregation are plotted in Figs. 5-9, and 10, and listed in Table V-III. Table V-III cone. (mg/ml) 5-31 316 1-90 0-975  E  (K) 214 186 166 138  ± 16-5 ± 220 ± 200 ±160  T  c  (K) 289-8 ± 1-4 295 1 ± 1-8 300-2 ± 1-6 333-4 ± 1-3  From Figs. 5-9 and 10 it appears,firstly,that the aggregation energy increases in a logarithmic fashion with concentration, and secondly, that the temperature of aggregation decreases exponentially with concentration. This is in good agreement with the results of Tremaine and Ronald. If we refer to Fig. 1-1 we can see as a consequence of the latter result aggregation commences at a lower temperature for higher concentrations. The first result implies there is a higher rate of change of the aggregation rate with temperature for higher concentrations. This is reflected in Fig. 1-1. As  CHAPTER  5: RESULTS  67  M E A N NUMBER O F PARTICLES PER A G G R E G A T E v s TIME 300-r  O  UJ  CH  o o <  250  X  CH  (/)  200  LJJ  H  _l  o r—  CH  O  or  150  100-  LU CD  LEGEND 50 H  < LU  300  600  900  1200  TIME s e c o n d s Figure 5-5 Theoretical fit by ANALYSIS to aggregation data.  A  THEORY  X  DATA  1500  1800  CHAPTER 5: RESULTS  MEAN NUMBER OF PARTICLES PER AGGREGATE v s TIME 120  LJ  y  1  I  TIME s e c o n d s Figure 5-6 Alternative fit by ANALYSIS to aggregation data.  CHAPTER 5: RESULTS  M E A N NUMBER OF PARTICLES P E R A G G R E G A T E v s TIME 40-i Ld  l  I  I  I  0  50  100  150  1  1  1  1  200 250 300 350  TIME s e c o n d s  Figure 5 - 7 Final fit by ANALYSIS to aggregation data.  [  400  5: RESULTS  CHAPTER  70  L O G O F T H E A G G R E G A T I O N RATE v s T E M P FOR F O U R DIFFERENT C O N C E N T R A T I O N S 0.12 -  i  0.11  • 0.10  or *  0.09  ® O  0.08  c o o  o  +  +  0.07  o  0.06  +  D  A  A  O  C O N C .  0.05  O  •  0.04  5.31  + 3.16 A  1.90  •  .975  0.03 0.02 25  30  35  40  45  50  TEMP celsius F i g u r e 5-8 Graph of 1/ l " ( , f c r ) versus T for the four concentrations. 3t  0  B  CHAPTER  5: RESULTS  AGGREGATION E N E R G Y v s C O N C . 250-  225X  200-  c  > >O  X  175-  CH  X  LxJ LU  150X  125-  100-  1  1  0  1  1  2  1  1  1  I  3  4  5  6  CONC. m g per m l Figure 5 - 9 Graph of energy of aggregation vs concentration.  7 1  CHAPTER  5: RESULTS  TEMP. O F AGGREGATION v s C O N C . 40  3 5 -  V) CD  o  3 0 -  o LJ  CH  2 5 -  O O < O  2 0  CL  15  H  10 ^ 1  2  3  4  5  CONC. mg per ml  Figure 5-10 Graph of temperature of aggregation vs concentration.  72  CHAPTER  5: RESULTS  each specimen was heated at a rate of 0-25C/min the start of each curve represents the commencement of measurable aggregation, and the near-vertical part of each curve corresponds to the temperature and time at which the virus is aggregating so rapidly that the solution has become turbid. The first result is reflected in the decrease in the time interval between these two stages that is apparent in Fig. 1-1. Measurements taken over long periods in the turbid regime show k continex  ually increasing, and it is assumed that if left for long enough at a temperature greater than T the virus would aggregate into one large mass. c  5.5 Suggestions for Further Study Further study could well be aimed at trying to understand the bonding process, through which the aggregation takes place, by varying the enviroment of the viruses. The following further suggestion for investigations were arrived at through consultation with Dr. Tremaine. (i) An investigation of the effect of various chemicals on the aggregation of CRSVN, particularly those that were studied by Tremaine and Ronald in their 5  absorbance studies . (ii) An investigation of the swelling of CRSV-N with pH adjustments, from pH 5 to pH 7, and the effects of divalent cations on the swelling. This might include 27  a study of any hysteresis effects similar to those reported by Tremaine and 28  Harrison . (iii) The size of twelve particle aggregates, and linked aggregates, are known in CRSV-A, as are their sedimentation coefficients. However, their non spherical shape means that one could investigate their diffusion coefficient.  73  C H A P T E R 6: C O N C L U S I O N S  CHAPTER 6 CONCLUSIONS  The aggregation rate of CRSV was measured for a range of temperatures and concentrations. The function exp\-E /k {T-T )} agg  B  c  if  T>T  C  if T < T  c  was found to fit the results well. It was also found that the aggregation parameters, E g and T , were a function of concentration. E ag  c  agg  increased with concentration.  However, the rate of increase decreases. T decreased with concentration. Similarly, c  the rate of decrease lessens. The size of the virus was confirmed to be 34 nm. The diffusion coefficient 2  was found to be 1-24 x 10~ cm /s and the molecular weight 8-3 x 10 daltons. The 7  6  results were in good accord with the work of Dr. Tremaine. The measurements on the Latex spheres were consistent with an expected experimental accuracy of Z\% in the hydrodynamic radius of the scattering particle.  BIBLIOGRAPHY  BIBLIOGRAPHY  1. A . A . B . Des. PI. Vir. 308, Sep (1985). 2. Smoluchowski, M . Von, Z. Phys. Chem. 92, 129 (1917). 3. Tremaine, J . H . , Ronald, W.P., and Valcic, A . , Phytopathology  66, 34 (1976).  4. Tremaine, J . H . , Ronald, W.P., and McGauley, A . M . , Physiol. Biochem. 74, 161 (1984). 5. Tremaine, J . H . , Ronald, W.P., Physiol. Biochem. 75, 467 (1985). 6. Cummins, H.Z., and Swinney, H . L . , Prog. Opt. 8, 135 (1970). 7. Cummins, H.Z., et al., Biophys. J . 9, 518 (1969). 8. Koppel, D . E . , J . Chem. Phys. 57, 4814 (1972). 9. Reichl, L . , A Modern Course in Statistical Physics, (University of Texas Press, 1980). 10. Jakeman, E . , and Pike, E . R . , J . Phys. A : Math., Nucl., Gen., 2, 411 (1969). 11. Koppel, D . E . , J . Appl. Phys. 42, 3216 (1971). 12. Berne, B . J . , and Pecora, R., Dynamic Light Scattering, (John Wiley and Sons, Inc., 1976). 13. Chandrasekhar, S., Rev. Mod. Phys. 15, 1 (1943). 14. Zsigmondy, R., Zeits. f. Physik. Chemie. 92, 600 (1917). 15. Freundlich, H . , Kapillar chemie II, 140 (1930-1932). 16. Jakeman, E . , Photon Correlation  and Light Beating Spectroscopy Ed: Cum-  mins, H.Z., and Pike, E . R . , Nato Advanced Studies Institute Series (Plenum Press, New York, 1975).  75  BIBLIOGRAPHY  17. Degiorgio, V . , Photon Correlation Spectroscopy and Velocimetry, Ed: Cummins, H.Z., and Pike, E . R . , Nato Advanced Studies Institute Series (Plenum Press, New York, 1976). 18. Foord, R., et ai, Nature 227, 242 (1970). 19. Jakeman, E . , and Pike, E.R., Advan. Quantum Elelctron 2, 1 (1974). 20. Pusey, P.N., et al., Biochem 13, 952 (1974). 21. Jakeman, E . , Pike, E.R., and Swain, S., J. Phys. A : Math., Nucl., Gen. 4, 517 (1971). 22. Instruction Manual, Type K7023 Malvern Digital Correlator. ni  23. C . R . C . Hnbk. Chem. Phys. 6 2 e d , F-42 (1982). 24. Seragen Diagnostics, Indianapolis, IN 46225, U.S.A. 25. Cummins, H.Z., Knable, N., and Yeh, Y . , Phys. Rev. Letts. 12, 150 (1964). 26. Hill, B . J . , Spragg, S.P., and McNeil-Watson, F . , Polymer 25, 467 (1984). 27. Tremaine, J.H., Mackenzie, D . J . , and Ronald, W.P., Virology 144, 80 (1985). 28. Harrison, S . C , Adv. Virus Res. 28, 117 1983  APPENDIX 1: EVALUATION  OF VELOCITY AUTOCORRELATION  FUNCTION  77  APPENDIX 1  EVALUATION OF VELOCITY AUTOCORRELATION  FUNCTION  If we consider the motion of the macromolecules in the solution to arise from incessant random bombardment by the solvent molecules, then the theories of Brownian motion can be used to calculate the integral of the velocity autocorrelation function r oo  /= /  (v {0)v {t)) dt x  x  Jo To do this one makes the observation that  r(t) = /  v(u) du.  (Al.l)  Jo From which it follows that,  r2  ( (0>=/  f (v{ui)v[u ))duidu 2  Jo Jo  (A1.2)  2l  If we denote the autocorrelation function of the velocity by the symbol  K (u\u ) v  2  and note the following properties: (i) In a stationary ensemble the function K (u\ ,u ) depends only on the time v  2  interval (ui - u ).Denoting this by the symbol s, we have 2  # i > ( « l , «i  + s) —  K  v{s)-  APPENDIX  1: EVALUATION  OF VELOCITY AUTOCORRELATION  FUNCTION  (ii) The qualityK [0), which is equal to the mean square value of the variable t ; v  at time uj must be positive definite, and, in a stationary ensemble, it must be constant. (iii) For any value of s, the magnitude of the function K (s) v  cannot exceed the  value at K {0). v  (iv) The function K [s) is symmetric about the value s = 0. v  (v) As s becomes large in comparison with the characteristic time T, the values of v ( u i ) and v(uj + s) become uncorrected, that is  lim K (s) - 0. v  To evaluate Eq. 1.2 then, make the following change of variables  S — -(ui + 2  u)  and  2  s  =  ( t t 2  - «i)  giving /=/  / •'O  K {s)dSds+  /  v  •'-25  J%  K (s)dSds. v  (A1.3)  J-2(t-S)  But in view of property (v) the function K (s) will only be non-zero in the narrow v  region of the order of T around the central value of s = 0. Therefore if t ~S> r the limits of integration for s may be replaced by — oo and +oo with the result +  00  (v{0)v{s)) de. /  (A1.4)  -oo  An established result of the Einstein and Langevin theories of Brownian Motion  8  is that for ( > r 2  (r ) = 6Dt.  (A1.5)  APPENDIX 1: EVALUATION OF VELOCITY AUTOCORRELATION FUNCTION 79 Equating Eq. A1.5 and Eq. A1.4 we arrive at  + oo  /  (v(0)v{s)) ds,  (A1.6)  (v {0)v [s)} ds.  (A1.7)  - oo  or •+oo r-t-oo  D = / -'0  x  x  APPENDIX  2: DESCRIPTION  OF THE CORRELATOR  CONTROLS  APPENDIX 2  DESCRIPTION OF T H ECORRELATOR CONTROLS  A diagram of the front panel of the correlator is illustrated in Fig. A l - 1 . A description of the controls is as follows.  Input Channels A and B The input channels have an input impedance of 5017. Logical 1 -1 volt Logical 0 0 volt Absolute maximum input levels +1 volt to -3 volts Pulse width 25ns Rise time not to exceed lfis Channels A and B have internal selection of positive or negative going triggering edges.  Function Switch Provides selection of the following modes of operation:Autocorrelation Single Clipped Autocorrelation Double Clipped Crosscorrelation Single Clipped Probability Density Probability Distribution  APPENDIX 2: DESCRIPTION  OF THE CORRELATOR  CONTROLS  Signal Averaging multiscaling Internal Trigger Signal Averaging multiscaling External Trigger  Figure A l - 1 Correlator front panel.  Clipping Level The decade switch determines the clip level. The clipped signal then being monitored on channel (01).  8 1  APPENDIX  2: DESCRIPTION  OF THE CORRELATOR  CONTROLS  Sample Time Three decade switches set up the sample time in steps of 0.05/xs up to a maximum of 9.95/zs. Another decade switch increases the sample time by multiples 5  of 10 up to the maximum sample time of 9.95 x 10 /xs. Number of Samples This decade switch determines the number of samples in multiples of 10. Up to 10 . 9  Readout Mode The contents of the dynamic store may be presented in four forms. These are as follows:Scope Here the presentation is on the C.R.T display whose X deflection represents the channel address, and whose Y deflection is proportionate to the the number of counts in the dynamic store. The Y outputs F.S.D. can be adjusted by means of the switch immediately below the readout mode selector switch, such that the Y sensitivity is variable from 50 counts per centimeter up to a maximum of 5 x l 0 (5M) 6  counts per centimeter.  Plot In this mode the Z80 microcomputer is used to load the data into a computer file *. Punch In this mode the START button is used to start the high speed punch FACIT Type 4070 75 CHAR/SEC. * see section 3.2  APPENDIX 2: DESCRIPTION  OF THE CORRELATOR  CONTROLS  Step In the plot position, with the Z80 disconnected the contents of each of the store channels can be stepped through and read from the display window.  Monitor Channel Overflow Indicators and Multipliers With the Monitor Channel Multipliers in the x l position the count capacity of the store of any monitor channel is 33,554,431. Any counts in excess of this number will cause the Monitor Channel Indicator for the particular channel in question to illuminate. The effective capacity for any channel can be extended by using the monitor channel multipliers up to a maximum count of 3,355,443,100. Monitor Channel 00 accumulates the total undipped counts in A. Monitor Channel 01 accumulates the total clipped counts in A. Monitor Channel 02 accumulates the total undipped counts in B. Monitor Channel 03 accumulates the total samples count.  Rate The signal from this socket is used to examine the pulse rate of either input channel.  83  APPENDIX  S: LISTING  OF  CORR.AG  APPENDIX 3  LISTING O F CORR.AG  EXTERNAL AUX C DATA TO BE READ IN FROM UNIT 5 REAL*4 STIME,THETA,TEMP INTEGER*4 NRUN,RCLIP,MULTS(4),NCOUNT,NCLIP INTEGER*4 ZERO,NSTIME,VALUES(100) C C VARIABLES FOR F I T T I N G A STRAIGHT LINE REAL*4 YF( 100) ,WTH00) ,E1 (3) ,E2(3) ,P(3) C VARIABLES FOR 2ND PLOT REAL*4 X I N T ( 1 0 0 ) , Y I N T ( 1 0 0 ) COMMON /PLOT2/ XINT,YINT,WT C C LOCAL VARIABLES REAL*4 X ( 1 0 0 ) , Y ( 1 0 0 ) , Y L O G ( 1 0 0 ) REAL*4 SM(100),FACTR,KD,Q,M2,RL,D20,VISC,DI LOGICAL*1 Y L A B E L ( 1 0 ) , E O F , F D ( 4 0 ) C COMMON M COMMON /PLTPAR/ NRUN,THETA,STIME,NSTIME,RCLIP,TEMP,MULTS, + XFIRST,XLAST,YFIRST,YFLAST,D,LABLEN,YLABEL C FACTR = 2.64911E05 CALL GFNAME(5,FD,LEN,& 5 0 0) WRITE(6,59) ( F D ( I ) , 1 = 1 , L E N ) 59 FORMAT(IX,'FILE = ',40A1) C C READ IN FIRST 5 LINES FROM UNIT 5 READ(5,50) NRUN,THETA,STIME,RCLIP,TEMP,MULTS,NCOUNT,NCLIP, + ZERO,NSTIME 50 FORMAT(I10,G10.4,G10.4,I2,G10.4,4I1,4(/,3X,19)) WRITE(6,60) NRUN,THETA,STIME,RCLIP,TEMP,MULTS,NCOUNT,NCLIP + ZERO,NSTIME 60 FORMAT(/, + 1X,' THE EXPERIMENTAL PARAMETERS ',/, + 1X,' RUN#=',I 8,5X,'THETA=',F7.3,11X,'STIME=',G10.4,/, + IX,'CLIPRATE=',I 2,11X,  APPENDIX  S: LISTING  OF  CORR.AG  + • TEMP=', G1 0. 4 , 1X, 'MULT.FACTS.=',412,/,/ + ,1X,'THE FIRST 4 CORRELATOR CHANNELS ARE ' ,/, + IX,' NCOUNT=',I9,4X,' NCLIP=',19,/, • 1X,' ZERO=',I9,4X,'NSTIME=',19) C C READ IN REST OF DATA N = 1 1 CONTINUE READ(5,51,END=100) NUM,VALUES(N) 51 F0RMAT(I3,I9) IF(NUM.LT.0)GO TO 100 N N + 1 GO TO 1 100 N = N - 1 C C EMPLOY MULTIPLICATION FACTORS NCOUNT - NCOUNT * 10.0**MULTS(1) NCLIP = NCLIP * 10.0**MULTS(2) ZERO « ZERO * 10.0**MULTS(3) NSTIME * NSTIME * 10.0**MULTS(4) C C CALCULATE THE VISCOSITY OF WATER VISC=(( 1 .3272M20-TEMP)) +-(0.001053*((TEMP-20.0)**2.0)))/(TEMP+l05) VISC«=1 .002E-02M 10.0**VISC) C C CALCULATE XN,BL AND CF XN « FLOAT(NCOUNT) / FLOAT(NSTIME) BL = XN * FLOAT(NCLIP) CF = (XN+1.0) / (FLOAT(RCLIP)+1.0) WRITE(6,61) XN,BL,CF 61 FORMAT(/,IX,'XN = NCOUNT/NSTIME =',T30,G10.4, + /,1X,'BL «= XN * NCLIP ,T30,G10.4, + /,1X,'CF « (XN+1 )/(CLIPRATE+1 ) «=' ,T30 ,G1 0. 4 , + /,1X,'B = EXPERIMENTAL PARAMETER (DARK NOISE)',/) C  C CALCULATE YMAX YMAX •= 0.0 DO 150 I* 1 ,N YMAX «= AMAX1 (YMAX, FLOAT (VALUES (I )) ) 150 CONTINUE YNORM •= YMAX-BL C C FILL X,Y AND YLOG ARRAYS WRITE(6,662) 662 FORMAT(IX, +'INPUT = DATA READ IN;',/, +1X,'X = MULTIPLE OF STIME;',/, + 1X/Y1 = G(T) = (INPUT - BL)/(YMAX - BL);',/, + 1X,'Y2 <= LN G(T) + LN(B/CF) = LN( INPUT/(BL - 1) ) ; ' , / , +1X,'WT = ( INPUT/(BL - 1) ) ** 2.0 = STAT WEIGHT ',/,/, +' INPUT',T20,'X',T29,'Y1 OR G(T)', +T42,'LN G(T) + LN(B/CF)',T65,'WT') DO 200 1=1,N X(I) = (1-1) * STIME Y(I) = (FLOAT(VALUES(I))-BL)/YNORM YLOG(I) = ALOG(FLOAT(VALUES(l))/BL - 1) SM(I)=FLOAT(VALUES(I))/(BL - l ) WT(I)=SM(I)**2.0 WRITE(6,62) VALUES(I),X(I),Y(I),YLOG(I),WT(I)  APPENDIX  62 200  S: LISTING OF CORR.AG  86  FORMAT(I10,4X,F10.3,4X,F10.8,4X,F10.7,8X,F10.7) CONTINUE  C C MAKE FIRST PICTURE CALL MOVEC(4 , ' G (T) ' ,YLABEL) LABLEN = 4 D «= 99999.99 CALL PICT(N,X,Y) EOF = .FALSE. C C FIT POLYNOMIAL ORDER 2 P(1) = 0.0 P(2) «= 0.0 P(3) = 0.0 WZ = 1.0 D = 99999.99 M = 3 NI = 6 EPS = 0.001 WRITE(6,64) 64 FORMAT(1X,'FITTING POLYNOMIAL : Y = PI + P2*X + P3*Xa2. ') CALL LQF(X,YLOG,YF,WT,E1,E2,P,WZ,N,M,NI,ND,EPS,AUX) WRITE(6,63) (El(I),1=1,M),ND 63 FORMAT(/,1X,'E1=',E12.6,2(4X,E12.6),6X,'ND=',14) WRITE(6,663) (E2(I),I=1,M) 663 FORMAT(1X,'E2=',E12.6,2(4X,E12.6),///) IF(ND.NE.1)GO TO 300 C RADS «= (3. 14159265 * THETA )/360. KD = FACTR * SIN(RADS) D=ABS(P(2))*1.0E+06/(2.0*KD**2) M2=P(3)*1.OE+12 RL=KD*KD*D Q=M2/(RL**2.0) DI = 1 ,0E+7*(1 .3807E-16*(273.16+TEMP))/(3.0*3.1415926*VISC*D) FRPAC=0.5 VNUM=(((DI/34.0)**3.0)-1)*FRPAC+1 .0 WRITE(6,66) 66 FORMAT(1X,'K=SCATTERING VECTOR (/cm) ' ,/, +1X,'D=DIFFUSION COEFICIENT (cm/sec)') WRITE(6,67) 67 FORMAT(1X,'RL=RAYLEIGH LINEWIDTH ( h z ) ' , / , +1X,*M2=2nd MOMENT OF LINE WIDTH DISTRIBUTION (hz**2)') WRITE(6,68) 68 FORMAT(IX,'Q=THE QUALITY FACTOR ( d i m e n s i o n l e s s ) ' , / , +1X,'DIA=THE AVERAGE DIAMETER OF THE PARTICLES (nm)') WRITE(6,69) 69 FORMAT(1X,'KBAR=THE MEAN NUMBER OF PARTICLES PER AGGREGATE') WRITE(6,65) KD,D,RL,M2,Q,DI,VNUM 65 FORMAT (1X,/,/,/, IX, 'K «= ' ,G 1 6. 6, 5X, ' D = ' ,G1 6. 6, 6X, ' RL = ' +.G16.6,/,1X,'M2 = ',G16.6,5X,'Q = ',G16.6,5X,'DIA = ',G16.6 +,/,IX,'KBAR - ',F11.2) C C MAKE SECOND PICTURE 300 CONTINUE CALL MOVEC(7,'LN G(T)',YLABEL) LABLEN = 7 CALL PICT(N,X,YLOG) C 500 STOP  APPENDIX  8: LISTING  OF  CORR.AG  END FUNCTION AUX(P,D,X,L) DIMENSION P(1),D(1) COMMON M D( 1 ) -1.0 AUX = P(1) DO 10 J = 2,M D(J) = D(J-1) * X AUX = AUX + P(J) * D(J) CONTINUE RETURN END SUBROUTINE PICT(N,X,Y) TYPE OF ARQUMENTS ***************** INTEGER* 4 N,NRUN,NSTIME,RCLIP,MULTS(4),LABLEN REAL*4 X(N),Y(N),THETA,STIME,TEMP,YFIRST,YFLAST,D,RVAR LOGICAL*1 YLABEL(10) COMMON /PLTPAR/ NRUN,THETA,STIME,NSTIME,RCLIP,TEMP,MULTS, + XFIRST,XLAST,YFIRST,YFLAST,D,LABLEN,YLABEL RANGE OF X AND Y AXES ********************* XMIN=X(1) XMAX=X(N) YMIN=99999.99 YMAX=-99999.99 DO 1 J=1,N YMIN=AMIN1(Y(J),YMIN) YMAX=AMAX1(Y(J),YMAX) CONTINUE EXTRA = (YMAX-YMIN)/20.0 YMAX = YMAX + EXTRA YMIN = YMIN - EXTRA PICTURE ORIGIN ************** XO=0.5 YO=0.5 AXES LENGTHS & SCALE FACTORS **************************** XLEN=11.0 YLEN=9.0 XSCALE=XLEN/(XMAX-XMIN) YSCALE=YLEN/(YMAX-YMIN) TICK MARKS ********** XTIC=STIME* * 2.0 YTIC=(YMAX-YMIN)/I 0.0 XGAP=XSCALE*XTIC YGAP=YSCALE*YTIC TIC-0.1 X-AXIS DRAWING  APPENDIX  S: LISTING  OF CORR.AG  CALL PLOT(XO YO,3) XX=XO REALN=XMIN BIT = 0.00001 NDIGIT=2 HT=0.1 XSHIFT=0.3 YSHIFT=0.2 CONTINUE CALL PLOT(XX,YO,2) CALL PLOT(XX,YO+TIC,2) CALL NUMBER(XX-XSHIFT,YO-YSHIFT,HT,REALN,0.0,NDIGIT) CALL PLOT(XX YO,3) REALN=REALN+XTIC IF(REALN.GT.XMAX+BIT)GO TO 15 XX=XX+XGAP GO TO 10 CALL PLOT(XLEN+XO,YO,2) HT=0.14 NLAB=7 XX=(XO+XLEN)/2.0 - (6/7 * HT * NLAB) CALL SYMBOL(XX,0.0,HT,'DELAY T',0.0,NLAB) f  f  Y-AXIS DRAWING ************** CALL PLpT(XO,YO,3) YY«=YO REALN=YMIN NDIGIT=3 HT=0.1 XSHIFT=0.6 YSHIFT=0.05 CONTINUE CALL PLOT(XO,YY,2) CALL PLOT(XO+TIC,YY,2) CALL NUMBER(XO-XSHIFT,YY-YSHIFT,HT REALN,0.0,NDIGIT) CALL PLOT(XO,YY,3) REALN=REALN+YTIC IF(REALN.GT.YMAX+BIT)GO TO 25 YY=YY+YGAP GO TO 20 CALL PLOT(XO,YLEN+YO,2) HT=0.14 NLAB=LABLEN YY=(YO+YLEN)/2.0 - (6/7 * HT * NLAB) XX=XO - XSHIFT - 0.1 CALL SYMBOL(XX,YY,HT,YLABEL,90.0,NLAB) IF(D.GT.99999.0)GO TO 29 r  XX • YY «= CALL XX = YY = CALL  FITTED POINTS ************* (XFIRST-XMIN)*XSCALE + XO (YFIRST-YMIN)*YSCALE + YO SYMBOL(XX,YY,0.14,0,0.0,-1) (XLAST-XMIN)*XSCALE + XO (YFLAST YMIN)*YSCALE + YO SYMBOL(XX,YY,0.14,0,0.1,-1) -  PLOTTING THE DATA  88  APPENDIX S: LISTING OF CORE.AG 8 9 29  30  NSYMB=3 HT=0.14 DO 30 1=1,N XX=(X(I)-XMIN)*XSCALE+XO YY=(Y(I )-YMIN)*YSCALE+YO CALL SYMBOL(XX,YY,HT,NSYMB,0.0,-1) CONTINUE  C C  PICTURE TITLE  Q  * * * * * * * * * * * * *  400  HT=0.14 XX=XO YY=YO+YLEN+2.5*HT  C CALL CALL RVAR CALL CALL CALL RVAR CALL CALL  SYMBOL(XX,YY,HT,'RUN#=',0.0,5) WHERE(XX,YIGNRE) * FLOAT(NRUN/100) NUMBER(XX+HT,YY,HT,RVAR,0.0,-1) WHERE(XX,YIGNRE) SYMBOL(XX,YY,HT,'-*,0.0, 1 ) = NRUN - IFIX(RVAR)* 100 WHERE(XX,YIGNRE) NUMBER(XX,YY,HT,RVAR,0.0,-1)  CALL CALL CALL CALL  WHERE(XX,YIGNRE) SYMBOL(XX+2.0*HT, YY,HT,' THETA«=' ,0.0,6) WHERE(XX,YIGNRE) NUMBER(XX+HT,YY,HT,THETA,0.0,2)  CALL CALL CALL CALL  WHERE(XX,YIGNRE) SYMBOL(XX+2.0*HT,YY,HT,'STIME=',0.0,6) WHERE(XX,YIGNRE) NUMBER(XX+HT,YY,HT,STIME,0.0,3)  C  C  C  299  CALL WHERE(XX,YIGNRE) CALL SYMBOL(XX+2.0*HT,YY,HT,'#STIMES=',0.0,8) CALL WHERE(XX,YIGNRE) RVAR = FLOAT(NSTIME/100) CALL NUMBER(XX+HT,YY.HT,RVAR,0.0,-1) RVAR = NSTIME - IFIX(RVAR)* 100 IF(RVAR.GE.10.0)GO TO 299 CALL WHERE(XX,YIGNRE) CALL SYMBOL(XX,YY,HT,'0*,0.0,1) CALL WHERE(XX,YIGNRE) CALL NUMBER(XX,YY,HT,RVAR,0.0,-1)  C CALL CALL CALL RVAR CALL  WHERE(XX,YIGNRE) SYMBOL(XX+2.0*HT,YY,HT,'CLIPRATE=',0.0,9) WHERE(XX,YIGNRE) «= FLOAT (RCLIP) NUMBER(XX+HT,YY,HT,RVAR,0.0,-1)  C XX = XO YY = YO + YLEN + HT C CALL SYMBOL(XX,YY,HT,'TEMP=',0.0,5) CALL WHERE(XX,YIGNRE) CALL NUMBER(XX+HT,YY,HT,TEMP,0.0,2) C CALL WHERE(XX,YIGNRE)  APPENDIX  301  S: LISTING OF CORR.AG  CALL SYMBOL(XX+2.0*HT,YY,HT,'MULTS=',0.0,6) DO 301 1=1,4 CALL WHERE(XX,YIGNRE) RVAR = FLOAT(MULTS(I)) CALL NUMBER(XX+HT,YY,HT,RVAR,0.0,-1) CONTINUE  C IF(D.GT.99999.0)GO TO 500 CALL WHERE(XX,YIGNRE) RVAR = D*10**9 CALL SYMBOL(XX+2.0*HT,YY,HT,'D=',0.0,2) CALL WHERE(XX,YIGNRE) CALL NUMBER(XX+HT,YY,HT,RVAR,0.0,3) CALL WHERE(XX,YIGNRE) CALL SYMBOL(XX+(HT/2.0),YY,HT,'X10**-09',0.0,8) C C 500  XX=XLEN+XO+2.0 CALL PLOT(XX,0.0,-3) RETURN END  90  

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